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# Integral eigenvalues of a matrix

Suppose $M$ is a $n \times n$ matrix such that, $$M=\left[\begin{array}{ccccccc} 0 & k_2&k_3&\cdots &k_{n-1}&k_n\\ k_{1} & 0 & k_3&\cdots &k_{n-1}&k_n\\ k_1&k_{2} & 0 &\cdots &k_{n-1}&k_n\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ k_{1} & k_{2}& k_3&\cdots & 0 &k_n\\ k_1&k_{2} & k_3&\cdots &k_{n-1}& 0\\ \end{array}\right]$$ with the characteristic polynomial, $0=|M-xI|=(-1)^n[x^n - (\displaystyle\sum_{\stackrel{i_1<i_2}{i_1,i_2=1}}^{n}{k_{i_1} k_{i_2}}) \ x^{(n-2)}- (2\displaystyle\sum_{\stackrel{i_1<i_2<i_3}{i_1,i_2,i_3=1}}^{n}{k_{i_1} k_{i_2}k_{i_3}}) \ x^{(n-3)} \ - \ ,\cdots, -(n-2)\displaystyle\sum_{\stackrel{i_1<i_2<i_3<...<i_{n-1}}{i_1,i_2,i_3,...,i_{n-1}=1}}^{n}{k_{i_1} k_{i_2}k_{i_3},...,k_{i_n}}\ x^{(1)}-(n-1)\displaystyle\sum_{\stackrel{i_1<i_2<i_3<...<i_n}{i_1,i_2,i_3,...,i_n=1}}^{n}{k_{i_1} k_{i_2}k_{i_3},...,k_{i_n}}\ x^{(0)} ].$
Then, how to prove that for $k_i\ne k_j$ for some $1\leq i,j\leq n$, the characteristic polynomial $|M-xI|=0$ has at least one roots which is not integer.
(Note that for $k_i = k_j$ for all $1\leq i,j\leq n$, the characteristic polynomial $|M-xI|=0$ has $n$ roots which are integers.)

It's not true. Any $M$ with exactly one nonzero column can serve as a counterexample. Counterexample exists even if all $k_i$s are nonzero and distinct. E.g. $M=\pmatrix{0&4\\ 1&0}$.
• you missed the condition that all the entries must be same in $M$ to get integral eigenvalues. – L S B. user255259 Sep 23 '15 at 7:41
• @LSB.user255259 What do you mean? Didn't you explicitly specified that the $k_i$s are not all the same? – user1551 Sep 23 '15 at 8:01
• If $M$ is $2 \times 2$ matrix with $k_i \times k_j$ is a perfect square then all the eigenvalues are integral. – L S B. user255259 Sep 23 '15 at 13:29
• @LSB.user255259 Yes. So this answers your question, doesn't it? You asked how to prove that $M$ has an non-integer eigenvalue when some $k_i$s are distinct, and I showed you a counterexample that all eigenvalues of $M$ can be integers even if all $k_i$s are distinct. – user1551 Sep 23 '15 at 13:45
• how to prove it for general $n \times n$. Do you mean that general $n \times n$ matrix for distinct $k_i$ have an integral eigenvalues? as in the case of $2 \times 2$ to get the integral eigenvalues, $k_i$ s must be a perfect square, then what must be the $k_i$s values for $n \times n$ matrix to get the integral eigenvalues. – L S B. user255259 Sep 23 '15 at 14:26


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# lap pace calculator

Help. 0.4 for 400 meters or 2.5 for 2.5Km). If you want... inverse\:laplace\:\frac{1}{x^{\frac{3}{2}}}, inverse\:laplace\:\frac{\sqrt{\pi}}{3x^{\frac{3}{2}}}, inverse\:laplace\:\frac{5}{4x^2+1}+\frac{3}{x^3}-5\frac{3}{2x}. ARENA ATTACK-RYAN CENTER Per Lap Pace Calculator Step 1 - Choose lap count from dropdown list. This calculator receives a series of lap times than outputs a total lap time, average lap time, best lap time, and longest lap time. Please try again using a different payment method. Each new topic we learn has symbols and problems we have never seen. More Cool Tools: Metric Duathlon Tool | Free Inverse Laplace Transform calculator - Find the inverse Laplace transforms of functions step-by-step This website uses cookies to ensure you get the best experience. This calculator was designed so that lap times can be entered quickly. Thanks for the feedback. This tool will help you calculate cumulative time, cumulative pace and cumulative average speed for splits taken at various intervals. Note- you don't need to enter hours or hundredths if they are zero (or if you didn't time yourself that precisely). This tool will help you calculate cumulative time, cumulative pace and cumulative average speed for splits taken at various intervals. By default the interval (dist column) is set to 1 - but you can change this suit your measurement (e.g. Although this calculator was designed for track and field races in mind, it can also be used for road races, cycling races, swim races, etc. Simply enter your goal time, race distance, and the your desired distance for splits. Brought to you by: Phil Barnes. Half Marathon = 78.49 5k = 18.55 Step 2 - Input goal time in the following format: "h:mm:ss". Triathlon Conversion Calculator | By default the interval (dist column) is set to 1 - but you can change this suit your measurement (e.g. Lap split calculator. For example if you want to swim 1000m in 15 minutes your pace would be 1:30 for each 100 Meters or 4Km per hour or 1:22 for each 100 yards or 2.5 Miles per hour The Swimming pace calculator can be used to calculate your swim pace in order to swim a given distance in a given time. By … Calculate the even splits for your goal time over a certain race distance and split/track length. Then, choose an option from the event drop-down list. Lap Time Calculator Random Passwords Generator. Cumulative Pace Calculator | If you have kilometer or mile splits, all you need to do is enter the corresponding number of hours, minutes, seconds and hundredths of a second for each split. To use the calculator, first enter the total time of your run. The lap times are accepted in Minutes and Seconds, and are to be entered in succession. This calculator uses Jack Daniels' formula for calculating "VDOT" (V̇ O 2 max) from a race performance, which is not the same thing as VO 2 max, but is somewhat related. Per Lap Pace. You can use VDOT to compare performances or to get training paces from (see Jack Daniels' calculator).. Imperial Duathlon Tool | The Official Runner's Cumulative Time and Pace Calculator, Back to the Cumulative Time and Pace Calculator. Message received. �9�� [Content_Types].xml �(� Ĕ�N�0��H�C�+j��Z�À#Lb��~qk���J'[�XZ��a6 iUiV9�X��,� �����M'�7���&�m0gE�s�T3������Jz�+�\���pϥ5LHC��&�'X����^I. Imperial Triathlon Tool | It is recommended to use a race distance between 1500 m and 50 km. 0.4 for 400 meters or 2.5 for 2.5Km). Step 3 - Hit the "Tab" key. Welcome to OnlineConversion.com. The unknowing... Learning math takes practice, lots of practice. To create your new password, just click the link in the email we sent you. Send me your comments and or suggestions: here. Metric Triathlon Tool | PK ! Use the calculator to figure out your pace per yard, mile, meter or kilometer, and view your splits in any of distance measurement. When you have entered all your data, click the 'Calculate' button on the last row of your data. Math can be an intimidating subject. The units (kilometers, meters, yards or miles) are not important - this tool will report your speed or pace in time/km and km/hr (if you enter your distance intervals in kilometers) or time/mile and mile/hr (if you enter your distance intervals in miles). The per lap pace required to hit your goal time will automatically be calculated in the "Per Lap Pace" column. This website uses cookies to ensure you get the best experience. Calculate average lap times. Et voilà! Just like running, it takes practice and dedication.

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## Existenzsätze für semilineare parabolische Systeme mit Funktionalen.(German)Zbl 0535.35038

Fakultät für Mathematik der Universität Karlsruhe. 65 S. (1982).
The author proves global existence results of the solutions of parabolic semilinear differential equations involving functionals, of the general form: $$L_ ku_ k=f_ k(t,x,u(\cdot),u_ x(\cdot))$$ in (0,T]$$\times \Omega$$, $$k=1,...,N$$, $$\Omega$$ open (bounded) domain in $${\mathbb{R}}^ n$$. Here $$L_ k$$ is a (non-autonomous) uniformly parabolic linear operator of the second order; the functional dependence on u, $$u_ x$$ of the functions $$f_ k$$ is, with respect to the time t, of ”delayed” type. The equations are supplemented by boundary conditions of the first or second type, possibly of non-linear type, and appropriate initial conditions. Existence is proved by considering the associated integral equations, making use of fixed point theorems of Banach, Schauder, Tychonoff. The author also outlines a number of models in applied sciences to which its results can be applied. The results extend those existing in the literature, even in the case in which the functional dependence is absent.
Reviewer: P.de Mottoni

### MSC:

 35K55 Nonlinear parabolic equations 35K10 Second-order parabolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions) 47H10 Fixed-point theorems


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What is the importance of air?

1 Answer
May 1, 2018

I'd say for the supply of oxygen...

Explanation:

Air is made up of around 78% of nitrogen gas $\left({N}_{2}\right)$, 21% of oxygen gas $\left({O}_{2}\right)$, and 1% of other gases, such as carbon dioxide, methane, noble gases, etc.

The most important gas for living organisms on Earth I'd say would be oxygen gas, due to it being essential for efficient energy production in the body, during aerobic respiration.

So, air is useful because it contains oxygen, which is what almost all, if not all organisms on Earth need to live and survive healthily.


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# Louis Poinsot

Louis Poinsot. Lithograph.

Louis Poinsot (1777–1859) was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a couple.

## Life

Everyone makes for himself a clear idea of the motion of a point, that is to say, of the motion of a corpuscle which one supposes to be infinitely small, and which one reduces by thought in some way to a mathematical point.
—Louis Poinsot, Théorie nouvelle de la rotation des corps (1834)

Louis was born in Paris on 3 January 1777. He attended the school of Lycée Louis-le-Grand for secondary preparatory education for entrance to the famous École Polytechnique. In October 1794, at age 17, he took the École Polytechnique entrance exam and failed the algebra section but was still accepted. A student there for two years, he left in 1797 to study at École des Ponts et Chaussées to become a civil engineer. Although now on course for the practical and secure professional study of civil engineering, he discovered his true passion, abstract mathematics.

Poinsot thus left the École des Ponts et Chaussées and civil engineering to become a mathematics teacher at the secondary school Lycée Bonaparte in Paris, from 1804 to 1809. From there he became inspector general of the Imperial University of France. He shared the post with another famous mathematician, Delambre. On 1 November 1809, Poinsot became assistant professor of analysis and mechanics at his old school the École Polytechnique. During this period of transitions between schools and work, Poinsot had remained active in research and published a number of works on geometry, mechanics and statics so that by 1809 he had an excellent reputation.

By 1812 Poinsot was no longer directly teaching at École Polytechnique using substitute teacher Reynaud, and later Cauchy, and lost his post in 1816 when they re-organized, but he did become admissions examiner and held that for another 10 years. He also worked at the famous Bureau des Longitudes from 1839 until his death. On the death of Joseph-Louis Lagrange in 1813, Poinsot was elected to fill his place at the Académie des Sciences. In 1840 he became a member of the superior council of public instruction. In 1846 he was awarded an Officer of the Legion of Honor, and on the formation of the Senate in 1852 he was chosen a member of that body. Poinsot was elected Fellow of the Royal Society of London in 1858. He died in Paris on 5 December 1859.

From the diary of Thomas Hirst, 20 December 1857:

...[Poinsot] shook me kindly by the hand, bid me be seated, and took his seat near me. He is now between 60 and 70 years old, with silver silken hair neatly arranged on a fine intelligent head. He is tall and thin, but although he now stoops with age and feebleness one can see that one time his figure was more than ordinarily graceful. He was loosely but neatly dressed in a large ample robe de chambre. His features are finely moulded — indeed everything about the man betokens good blood. He talks incessantly and well. I did not misunderstand a word, although he spoke always in a low tone, and now and then his voice dropped as if from weariness, but he never wandered from his point...

## Legacy and tributes

The crater Poinsot on the moon is named after Poinsot. A street in Paris is called Rue Poinsot (14th Arrondissement). Gustave Eiffel included Poinsot among the 72 names of prominent French scientists on plaques around the first stage of the Eiffel Tower.

## Work

"Poinsot was determined to publish only fully developed results and to present them with clarity and elegance. Consequently he left a rather limited body of work ..."
—Dictionary of Scientific Biography (see Sources)

Works include:

• Eléments de statique (1803) (translated as The Elements of Statics in 1848 by Thomas Sutton)1
• memoirs that dealt with the composition of moments and the composition of areas (1806)
• the general theory of equilibrium and of movements in systems (1806)
• polygons and polyhedra (1809)
• Theorie nouvelle de la rotation des corps (1834)

Poinsot was the inventor of geometrical mechanics, which showed how a system of forces acting on a rigid body could be resolved into a single force and a couple. Previous work done on the motion of a rigid body had been purely analytical with no visualization of the motion, and the great value of the work, as Poinsot says, it enables us to represent to ourselves the motion of a rigid body as clearly that as a moving point (Encyclopædia Britannica, 1911). In particular he devised, what is now known as, Poinsot's construction. This construction describes the motion of the angular velocity vector $\mathbf{\omega}$ of a rigid body with one point fixed (usually its center of mass). He proved that the endpoint of the vector $\mathbf{\omega}$ moves in a plane perpendicular to the angular momentum (in absolute space) of the rigid body.2

He discovered the four Kepler-Poinsot polyhedra in 1809. Two of these had already appeared in Kepler's work of 1619, although Poinsot was unaware of this. The other two are the great icosahedron and great dodecahedron, which some people call these two the Poinsot solids. In 1810 Cauchy proved, using Poinsot's definition of regular, that the enumeration of regular star polyhedra is complete.

Poinsot worked on number theory studying Diophantine equations. However he is best known for his work in geometry and, together with Monge, regained geometry's leading role in mathematical research in France in the 19th century. Poinsot also contributed to the importance of geometry by creating a chair of advanced geometry at the Sorbonne in 1846. Poinsot created the chair for Chasles which he occupied until his death in 1880.

## References

1. ^
2. ^ E.T. Whittaker, Analytic Dynamics of Particles and Rigid Bodies, Cambridge UP, 4th edition, (1938), p. 152 ff.

## Sources

• Bertrand, J. L. F. Discours aux funérailles de Poinsot, Paris, 1860.
• Bertrand, J. L. F. Notice sur Louis Poinsot, Journal des savants (1872), pp. 405–420.
• Taton, René (1970–80). "Poinsot, Louis". Dictionary of Scientific Biography 11. New York: Charles Scribner's Sons. pp. 61–62. ISBN 0684101149.

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# Spinor

A spinor visualized as a vector pointing along the Möbius band, exhibiting a sign inversion when the circle (the "physical system") is rotated through a full turn of 360°.[nb 1]

In geometry and physics, spinors are elements of a (complex) vector space that can be associated with Euclidean space.[nb 2] Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation.[nb 3] When a sequence of such small rotations is composed (integrated) to form an overall final rotation, however, the resulting spinor transformation depends on which sequence of small rotations was used, unlike for vectors and tensors. A spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360° (see picture), and it is this property that characterizes spinors. It is also possible to associate a substantially similar notion of spinor to Minkowski space in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913.[1][2] In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles.[nb 4]

Spinors are characterized by the specific way in which they behave under rotations. They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved (by a continuous path in the rotation group). There are two topologically distinguishable classes (homotopy classes) of paths through rotations that result in the same overall rotation, as famously illustrated by the belt trick puzzle (below). These two inequivalent classes yield spinor transformations of opposite sign. The spin group is the group of all rotations keeping track of the class.[nb 5] It doubly-covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors by definition is equipped with a (complex) linear representation of the spin group, meaning that elements of the spin group act as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class.[nb 6]

Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra. The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest to work with.[nb 7] After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations. The spinors are the column vectors on which these matrices act. In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices,[nb 8] and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, and hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way. As a representation of the spin group, this realization of spinors as (complex[nb 9]) column vectors will either be irreducible if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even.[nb 10]

## Introduction

A gradual rotation can be visualized as a ribbon in space (the TNB frame of the ribbon defines a rotation continuously for each value of the arc length parameter). Two gradual rotations with different classes, one through 2π and one through 4π, are illustrated here in the belt trick puzzle. A solution of the puzzle is a (continuous) manipulation of the belt, fixing the endpoints, that untwists it. This is impossible with the 2π rotation, but possible with the 4π rotation. A solution, shown in the second animation, actually gives an explicit homotopy in the rotation group between the 4π rotation and the trivial (identity) rotation.

What characterizes spinors and distinguishes them from geometric vectors and other tensors is subtle. Consider applying a rotation to the coordinates of a system. No object in the system itself has moved, only the coordinates have, so there will always be a compensating change in those coordinate values when applied to any object of the system. Geometrical vectors, for example, have components that will undergo the same rotation as the coordinates. More broadly, any tensor associated with the system (for instance, the stress of some medium) also has coordinate descriptions that adjust to compensate for changes to the coordinate system itself. Spinors do not appear at this level of the description of a physical system, when one is concerned only with the properties of a single isolated rotation of the coordinates. Rather, spinors appear when we imagine that instead of a single rotation, the coordinate system is gradually (continuously) rotated between some initial and final configuration. For any of the familiar and intuitive ("tensorial") quantities associated with the system, the transformation law does not depend on the precise details of how the coordinates arrived at their final configuration. Spinors, on the other hand, are constructed in such a way that makes them sensitive to how the gradual rotation of the coordinates arrived there: they exhibit path-dependence. It turns out that, for any final configuration of the coordinates, there are actually two ("topologically") inequivalent gradual (continuous) rotations of the coordinate system that result in this same configuration. This ambiguity is called the homotopy class of the gradual rotation. The belt trick puzzle (shown) famously demonstrates two different rotations, one through an angle of 2π and the other through an angle of 4π, having the same final configurations but different classes. Spinors actually exhibit a sign-reversal that genuinely depends on this homotopy class. This distinguishes them from vectors and other tensors, none of which can feel the class.

Spinors can be exhibited as concrete objects using a choice of Cartesian coordinates. In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of Pauli spin matrices corresponding to (angular momenta about) the three coordinate axes. These are 2×2 matrices with complex entries, and the two-component complex column vectors on which these matrices act by matrix multiplication are the spinors. In this case, the spin group is isomorphic to the group of 2×2 unitary matrices with determinant one, which naturally sits inside the matrix algebra. This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves,[nb 11] realizing it as a group of rotations among them,[nb 12] but it also acts on the column vectors (that is, the spinors).

More generally, a Clifford algebra can be constructed from any vector space V equipped with a (nondegenerate) quadratic form, such as Euclidean space with its standard dot product or Minkowski space with its standard Lorentz metric. Given a suitably normalized basis of V, the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations, and the space of spinors is the space of column vectors with $2^{\lfloor\dim V/2\rfloor}$ components on which those matrices act. Although the Clifford algebra can be defined abstractly in a coordinate-independent way, its particular realization as a specific algebra of matrices depends on which orthogonal axes the gamma matrices represent. So what precisely constitutes a "column vector" (or spinor) also depends on such arbitrary choices.[nb 13] The orthogonal Lie algebra (i.e., the infinitesimal "rotations") and the spin group associated to the quadratic form are both (canonically) contained in the Clifford algebra, so every Clifford algebra representation also defines a representation of the Lie algebra and the spin group.[nb 14] Depending on the dimension and metric signature, this realization of spinors as column vectors may be irreducible or it may decompose into a pair of so-called "half-spin" or Weyl representations.[nb 15]

## Overview

There are essentially two frameworks for viewing the notion of a spinor.

One is representation theoretic. In this point of view, one knows beforehand that there are some representations of the Lie algebra of the orthogonal group that cannot be formed by the usual tensor constructions. These missing representations are then labeled the spin representations, and their constituents spinors. In this view, a spinor must belong to a representation of the double cover of the rotation group SO(n, R), or more generally of double cover of the generalized special orthogonal group SO+(p, q, R) on spaces with metric signature (p, q). These double covers are Lie groups, called the spin groups Spin(n) or Spin(p, q). All the properties of spinors, and their applications and derived objects, are manifested first in the spin group. Representations of the double covers of these groups yield projective representations of the groups themselves, which do not meet the full definition of a representation.

The other point of view is geometrical. One can explicitly construct the spinors, and then examine how they behave under the action of the relevant Lie groups. This latter approach has the advantage of providing a concrete and elementary description of what a spinor is. However, such a description becomes unwieldy when complicated properties of spinors, such as Fierz identities, are needed.

### Clifford algebras

For more details on this topic, see Clifford algebra.

The language of Clifford algebras[3] (sometimes called geometric algebras) provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations, via the classification of Clifford algebras. It largely removes the need for ad hoc constructions.

In detail, let V be a finite-dimensional complex vector space with nondegenerate bilinear form g. The Clifford algebra Cℓ(V, g) is the algebra generated by V along with the anticommutation relation xy + yx = 2g(x, y). It is an abstract version of the algebra generated by the gamma or Pauli matrices. If V = Cn, with the standard form g(x, y) = xty = x1y1 + ... + xnyn we denote the Clifford algebra by Cℓn(C). Since by the choice of an orthonormal basis every complex vectorspace with non-degenerate form is isomorphic to this standard example, this notation is abused more generally if dimC(V) = n. If n = 2k is even, Cℓn(C) is isomorphic as an algebra (in a non-unique way) to the algebra Mat(2k, C) of 2k × 2k complex matrices (by the Artin-Wedderburn theorem and the easy to prove fact that the Clifford algebra is central simple). If n = 2k + 1 is odd, Cℓ2k+1(C) is isomorphic to the algebra Mat(2k, C) ⊕ Mat(2k, C) of two copies of the 2k × 2k complex matrices. Therefore, in either case Cℓ(V, g) has a unique (up to isomorphism) irreducible representation (also called simple Clifford module), commonly denoted by Δ, of dimension 2[n/2]. Since the Lie algebra so(V, g) is embedded as a Lie subalgebra in Cℓ(V, g) equipped with the Clifford algebra commutator as Lie bracket, the space Δ is also a Lie algebra representation of so(V, g) called a spin representation. If n is odd, this Lie algebra representation is irreducible. If n is even, it splits further into two irreducible representations Δ = Δ+ ⊕ Δ called the Weyl or half-spin representations.

Irreducible representations over the reals in the case when V is a real vector space are much more intricate, and the reader is referred to the Clifford algebra article for more details.

### Spin groups

The spin representation Δ is a vector space equipped with a representation of the spin group that does not factor through a representation of the (special) orthogonal group.

Spinors form a vector space, usually over the complex numbers, equipped with a linear group representation of the spin group that does not factor through a representation of the group of rotations (see diagram). The spin group is the group of rotations keeping track of the homotopy class. Spinors are needed to encode basic information about the topology of the group of rotations because that group is not simply connected, but the simply connected spin group is its double cover. So for every rotation there are two elements of the spin group that represent it. Geometric vectors and other tensors cannot feel the difference between these two elements, but they produce opposite signs when they affect any spinor under the representation. Thinking of the elements of the spin group as homotopy classes of one-parameter families of rotations, each rotation is represented by two distinct homotopy classes of paths to the identity. If a one-parameter family of rotations is visualized as a ribbon in space, with the arc length parameter of that ribbon being the parameter (its tangent, normal, binormal frame actually gives the rotation), then these two distinct homotopy classes are visualized in the two states of the belt trick puzzle (above). The space of spinors is an auxiliary vector space that can be constructed explicitly in coordinates, but ultimately only exists up to isomorphism in that there is no "natural" construction of them that does not rely on arbitrary choices such as coordinate systems. A notion of spinors can be associated, as such an auxiliary mathematical object, with any vector space equipped with a quadratic form such as Euclidean space with its standard dot product, or Minkowski space with its Lorentz metric. In the latter case, the "rotations" include the Lorentz boosts, but otherwise the theory is substantially similar.

### Terminology in physics

The most typical type of spinor, the Dirac spinor,[4] is an element of the fundamental representation of Cℓp+q(C), the complexification of the Clifford algebra Cℓp, q(R), into which the spin group Spin(p, q) may be embedded. On a 2k- or 2k+1-dimensional space a Dirac spinor may be represented as a vector of 2k complex numbers. (See Special unitary group.) In even dimensions, this representation is reducible when taken as a representation of Spin(p, q) and may be decomposed into two: the left-handed and right-handed Weyl spinor[5] representations. In addition, sometimes the non-complexified version of Cℓp,q(R) has a smaller real representation, the Majorana spinor representation.[6] If this happens in an even dimension, the Majorana spinor representation will sometimes decompose into two Majorana–Weyl spinor representations. Dirac and Weyl spinors are complex representations while Majorana spinors are real representations.

The Dirac, Lorentz, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra.[7]

Massive particles, such as electrons, are described as Dirac spinors. The classical neutrino of the standard model of particle physics is an example of a Weyl spinor. However, because of observed neutrino oscillation, it is now believed that they are not Weyl spinors, but perhaps instead Majorana spinors.[8] It is not known whether (spin-1/2) Weyl spinors exist in nature. In 2015, an international team led by Princeton University scientists announced that they had found a quasiparticle that behaves as a Weyl fermion.[9]

### Spinors in representation theory

Main article: Spin representation

One major mathematical application of the construction of spinors is to make possible the explicit construction of linear representations of the Lie algebras of the special orthogonal groups, and consequently spinor representations of the groups themselves. At a more profound level, spinors have been found to be at the heart of approaches to the Atiyah–Singer index theorem, and to provide constructions in particular for discrete series representations of semisimple groups.

The spin representations of the special orthogonal Lie algebras are distinguished from the tensor representations given by Weyl's construction by the weights. Whereas the weights of the tensor representations are integer linear combinations of the roots of the Lie algebra, those of the spin representations are half-integer linear combinations thereof. Explicit details can be found in the spin representation article.

### Attempts at intuitive understanding

The spinor can be described, in simple terms, as “vectors of a space the transformations of which are related in a particular way to rotations in physical space”.[10] Stated differently:[2]

Spinors […] provide a linear representation of the group of rotations in a space with any number $n$ of dimensions, each spinor having $2^\nu$ components where $n = 2\nu+1$ or $2\nu$.

Several ways of illustrating everyday analogies have been formulated in terms of the plate trick, tangloids and other examples of orientation entanglement.

Nonetheless, the concept is generally considered notoriously difficult to understand, as illustrated by Michael Atiyah's statement that is recounted by Dirac's biographer Graham Farmelo:[11]

No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the “square root” of geometry and, just as understanding the square root of −1 took centuries, the same might be true of spinors.

## History

The most general mathematical form of spinors was discovered by Élie Cartan in 1913.[12] The word "spinor" was coined by Paul Ehrenfest in his work on quantum physics.[13]

Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced his spin matrices.[14] The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group.[15] By the 1930s, Dirac, Piet Hein and others at the Niels Bohr Institute (then known as the Institute for Theoretical Physics of the University of Copenhagen) created toys such as Tangloids to teach and model the calculus of spinors.

Spinor spaces were represented as left ideals of a matrix algebra in 1930, by G. Juvet[16] and by Fritz Sauter.[17][18] More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero. In this manner the spinor space became a minimal left ideal in Mat(2, C).[19][20]

In 1947 Marcel Riesz constructed spinor spaces as elements of a minimal left ideal of Clifford algebras. In 1966/1967, David Hestenes[21][22] replaced spinor spaces by the even subalgebra Cℓ01,3(R) of the spacetime algebra Cℓ1,3(R).[18][20] As of the 1980s, the theoretical physics group at Birkbeck College around David Bohm and Basil Hiley has been developing algebraic approaches to quantum theory that build on Sauter and Riesz' identification of spinors with minimal left ideals.

## Examples

Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra Cℓp, q(R). This is an algebra built up from an orthonormal basis of n = p + q mutually orthogonal vectors under addition and multiplication, p of which have norm +1 and q of which have norm −1, with the product rule for the basis vectors

$e_i e_j = \Bigg\{ \begin{matrix} +1 & i=j, \, i \in (1 \ldots p) \\ -1 & i=j, \, i \in (p+1 \ldots n) \\ - e_j e_i & i \not = j. \end{matrix}$

### Two dimensions

The Clifford algebra Cℓ2,0(R) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, σ1 and σ2, and one unit pseudoscalar i = σ1σ2. From the definitions above, it is evident that (σ1)2 = (σ2)2 = 1, and (σ1σ2)(σ1σ2) = −σ1σ1σ2σ2 = −1.

The even subalgebra Cℓ02,0(R), spanned by even-graded basis elements of Cℓ2,0(R), determines the space of spinors via its representations. It is made up of real linear combinations of 1 and σ1σ2. As a real algebra, Cℓ02,0(R) is isomorphic to field of complex numbers C. As a result, it admits a conjugation operation (analogous to complex conjugation), sometimes called the reverse of a Clifford element, defined by

$(a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1\,$.

which, by the Clifford relations, can be written

$(a+b\sigma_1\sigma_2)^* = a+b\sigma_2\sigma_1 = a-b\sigma_1\sigma_2\,$.

The action of an even Clifford element γ ∈ Cℓ02,0(R) on vectors, regarded as 1-graded elements of Cℓ2,0(R), is determined by mapping a general vector u = a1σ1 + a2σ2 to the vector

$\gamma(u) = \gamma u \gamma^*\,$,

where γ is the conjugate of γ, and the product is Clifford multiplication. In this situation, a spinor[23] is an ordinary complex number. The action of γ on a spinor φ is given by ordinary complex multiplication:

$\gamma(\phi) = \gamma\phi$.

An important feature of this definition is the distinction between ordinary vectors and spinors, manifested in how the even-graded elements act on each of them in different ways. In general, a quick check of the Clifford relations reveals that even-graded elements conjugate-commute with ordinary vectors:

$\gamma(u) = \gamma u \gamma^* = \gamma^2 u\,$.

On the other hand, comparing with the action on spinors γ(φ) = γφ, γ on ordinary vectors acts as the square of its action on spinors.

Consider, for example, the implication this has for plane rotations. Rotating a vector through an angle of θ corresponds to γ2 = exp(θ σ1σ2), so that the corresponding action on spinors is via γ = ± exp(θ σ1σ2/2). In general, because of logarithmic branching, it is impossible to choose a sign in a consistent way. Thus the representation of plane rotations on spinors is two-valued.

In applications of spinors in two dimensions, it is common to exploit the fact that the algebra of even-graded elements (that is just the ring of complex numbers) is identical to the space of spinors. So, by abuse of language, the two are often conflated. One may then talk about "the action of a spinor on a vector." In a general setting, such statements are meaningless. But in dimensions 2 and 3 (as applied, for example, to computer graphics) they make sense.

Examples
$\gamma = \tfrac{1}{\sqrt{2}} (1 - \sigma_1 \sigma_2) \,$
corresponds to a vector rotation of 90° from σ1 around towards σ2, which can be checked by confirming that
$\tfrac{1}{2} (1 - \sigma_1 \sigma_2) \, \{a_1\sigma_1+a_2\sigma_2\} \, (1 - \sigma_2 \sigma_1) = a_1\sigma_2 - a_2\sigma_1 \,$
It corresponds to a spinor rotation of only 45°, however:
$\tfrac{1}{\sqrt{2}} (1 - \sigma_1 \sigma_2) \, \{a_1+a_2\sigma_1\sigma_2\}= \frac{a_1+a_2}{\sqrt{2}} + \frac{-a_1+a_2}{\sqrt{2}}\sigma_1\sigma_2$
• Similarly the even-graded element γ = −σ1σ2 corresponds to a vector rotation of 180°:
$(- \sigma_1 \sigma_2) \, \{a_1\sigma_1 + a_2\sigma_2\} \, (- \sigma_2 \sigma_1) = - a_1\sigma_1 -a_2\sigma_2 \,$
but a spinor rotation of only 90°:
$(- \sigma_1 \sigma_2) \, \{a_1 + a_2\sigma_1\sigma_2\} =a_2 - a_1\sigma_1\sigma_2$
• Continuing on further, the even-graded element γ = −1 corresponds to a vector rotation of 360°:
$(-1) \, \{a_1\sigma_1+a_2\sigma_2\} \, (-1) = a_1\sigma_1+a_2\sigma_2 \,$
but a spinor rotation of 180°.

### Three dimensions

Main articles Spinors in three dimensions, Quaternions and spatial rotation

The Clifford algebra Cℓ3,0(R) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, σ1, σ2 and σ3, the three unit bivectors σ1σ2, σ2σ3, σ3σ1 and the pseudoscalar i = σ1σ2σ3. It is straightforward to show that (σ1)2 = (σ2)2 = (σ3)2 = 1, and (σ1σ2)2 = (σ2σ3)2 = (σ3σ1)2 = (σ1σ2σ3)2 = −1.

$u^{\prime} = \rho^{(1/2)} u \rho^{(1/2)} = \rho u,$

and vector rotations

$u^{\prime} = \gamma \, u \, \gamma^*,$

where

$\left.\begin{array}{rcl} \gamma & = & \cos(\theta/2) - \{a_1 \sigma_2\sigma_3 + a_2 \sigma_3\sigma_1 + a_3 \sigma_1\sigma_2\} \sin(\theta/2) \\ & = & \cos(\theta/2) - i \{a_1 \sigma_1 + a_2 \sigma_2 + a_3 \sigma_3\} \sin(\theta/2) \\ & = & \cos(\theta/2) - i v \sin(\theta/2) \end{array}\right\}$ (1)

corresponds to a vector rotation through an angle θ about an axis defined by a unit vector v = a1σ1 + a2σ2 + a3σ3.

As a special case, it is easy to see that, if v = σ3, this reproduces the σ1σ2 rotation considered in the previous section; and that such rotation leaves the coefficients of vectors in the σ3 direction invariant, since

$(\cos(\theta/2) - i \sigma_3 \sin(\theta/2)) \, \sigma_3 \, (\cos(\theta/2) + i \sigma_3 \sin(\theta/2)) = (\cos^2(\theta/2) + \sin^2(\theta/2)) \, \sigma_3 = \sigma_3.$

The bivectors σ2σ3, σ3σ1 and σ1σ2 are in fact Hamilton's quaternions i, j and k, discovered in 1843:

$\begin{matrix}\mathbf{i} = -\sigma_2 \sigma_3 = -i \sigma_1 \\ \mathbf{j} = -\sigma_3 \sigma_1 = -i \sigma_2 \\ \mathbf{k} = -\sigma_1 \sigma_2 = -i \sigma_3. \end{matrix}$

With the identification of the even-graded elements with the algebra H of quaternions, as in the case of two dimensions the only representation of the algebra of even-graded elements is on itself.[24] Thus the (real[25]) spinors in three-dimensions are quaternions, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication.

Note that the expression (1) for a vector rotation through an angle θ, the angle appearing in γ was halved. Thus the spinor rotation γ(ψ) = γψ (ordinary quaternionic multiplication) will rotate the spinor ψ through an angle one-half the measure of the angle of the corresponding vector rotation. Once again, the problem of lifting a vector rotation to a spinor rotation is two-valued: the expression (1) with (180° + θ/2) in place of θ/2 will produce the same vector rotation, but the negative of the spinor rotation.

The spinor/quaternion representation of rotations in 3D is becoming increasingly prevalent in computer geometry and other applications, because of the notable brevity of the corresponding spin matrix, and the simplicity with which they can be multiplied together to calculate the combined effect of successive rotations about different axes.

## Explicit constructions

A space of spinors can be constructed explicitly with concrete and abstract constructions. The equivalence of these constructions are a consequence of the uniqueness of the spinor representation of the complex Clifford algebra. For a complete example in dimension 3, see spinors in three dimensions.

### Component spinors

Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra Cℓ(V, g) can be defined as follows. Choose an orthonormal basis e1en for V i.e. g(eμeν) = ημν where ημμ = ±1 and ημν = 0 for μν. Let k = ⌊ n/2 ⌋. Fix a set of 2k × 2k matrices γ1γn such that γμγν + γνγμ = 2ημν1 (i.e. fix a convention for the gamma matrices). Then the assignment eμγμ extends uniquely to an algebra homomorphism Cℓ(V, g) → Mat(2k, C) by sending the monomial eμ1eμk in the Clifford algebra to the product γμ1γμk of matrices and extending linearly. The space Δ = C2k on which the gamma matrices act is a now a space of spinors. One needs to construct such matrices explicitly, however. In dimension 3, defining the gamma matrices to be the Pauli sigma matrices gives rise to the familiar two component spinors used in non relativistic quantum mechanics. Likewise using the 4 × 4 Dirac gamma matrices gives rise to the 4 component Dirac spinors used in 3+1 dimensional relativistic quantum field theory. In general, in order to define gamma matrices of the required kind, one can use the Weyl–Brauer matrices.

In this construction the representation of the Clifford algebra Cℓ(V, g), the Lie algebra so(V, g), and the Spin group Spin(V, g), all depend on the choice of the orthonormal basis and the choice of the gamma matrices. This can cause confusion over conventions, but invariants like traces are independent of choices. In particular, all physically observable quantities must be independent of such choices. In this construction a spinor can be represented as a vector of 2k complex numbers and is denoted with spinor indices (usually α, β, γ). In the physics literature, abstract spinor indices are often used to denote spinors even when an abstract spinor construction is used.

### Abstract spinors

There are at least two different, but essentially equivalent, ways to define spinors abstractly. One approach seeks to identify the minimal ideals for the left action of Cℓ(V, g) on itself. These are subspaces of the Clifford algebra of the form Cℓ(V, g)ω, admitting the evident action of Cℓ(V, g) by left-multiplication: c : cxω. There are two variations on this theme: one can either find a primitive element ω that is a nilpotent element of the Clifford algebra, or one that is an idempotent. The construction via nilpotent elements is more fundamental in the sense that an idempotent may then be produced from it.[26] In this way, the spinor representations are identified with certain subspaces of the Clifford algebra itself. The second approach is to construct a vector space using a distinguished subspace of V, and then specify the action of the Clifford algebra externally to that vector space.

In either approach, the fundamental notion is that of an isotropic subspace W. Each construction depends on an initial freedom in choosing this subspace. In physical terms, this corresponds to the fact that there is no measurement protocol that can specify a basis of the spin space, even if a preferred basis of V is given.

As above, we let (V, g) be an n-dimensional complex vector space equipped with a nondegenerate bilinear form. If V is a real vector space, then we replace V by its complexification V ⊗RC and let g denote the induced bilinear form on V ⊗RC. Let W be a maximal isotropic subspace, i.e. a maximal subspace of V such that g|W = 0. If n =  2k is even, then let W be an isotropic subspace complementary to W. If n =  2k + 1 is odd, let W be a maximal isotropic subspace with W ∩ W = 0, and let U be the orthogonal complement of W ⊕ W. In both the even- and odd-dimensional cases W and W have dimension k. In the odd-dimensional case, U is one-dimensional, spanned by a unit vector u.

### Minimal ideals

Since W′ is isotropic, multiplication of elements of W′ inside Cℓ(V, g) is skew. Hence vectors in W′ anti-commute, and Cℓ(W′, g|W) = Cℓ(W′, 0) is just the exterior algebra ΛW′. Consequently, the k-fold product of W′ with itself, Wk, is one-dimensional. Let ω be a generator of Wk. In terms of a basis w1,..., wk of in W′, one possibility is to set

$\omega=w'_1w'_2\cdots w'_k.$

Note that ω2 = 0 (i.e., ω is nilpotent of order 2), and moreover, wω = 0 for all w′ ∈ W. The following facts can be proven easily:

1. If n = 2k, then the left ideal Δ = Cℓ(V, g)ω is a minimal left ideal. Furthermore, this splits into the two spin spaces Δ+ = Cℓevenω and Δ = Cℓoddω on restriction to the action of the even Clifford algebra.
2. If n = 2k + 1, then the action of the unit vector u on the left ideal Cℓ(V, g)ω decomposes the space into a pair of isomorphic irreducible eigenspaces (both denoted by Δ), corresponding to the respective eigenvalues +1 and −1.

In detail, suppose for instance that n is even. Suppose that I is a non-zero left ideal contained in Cℓ(V, g)ω. We shall show that I must be equal to Cℓ(V, g)ω by proving that it contains a nonzero scalar multiple of ω.

Fix a basis wi of W and a complementary basis wi′ of W′ so that

wiwj′ +wjwi = δij, and
(wi)2 = 0, (wi′)2 = 0.

Note that any element of I must have the form αω, by virtue of our assumption that I ⊂ Cℓ(V, g) ω. Let αωI be any such element. Using the chosen basis, we may write

$\alpha = \sum_{i_1

where the ai1…ip are scalars, and the Bj are auxiliary elements of the Clifford algebra. Observe now that the product

$\alpha\omega = \sum_{i_1

Pick any nonzero monomial a in the expansion of α with maximal homogeneous degree in the elements wi:

$a = a_{i_1\dots i_{max}}w_{i_1}\dots w_{i_{max}}$ (no summation implied),

then

$w'_{i_{max}}\cdots w'_{i_1}\alpha\omega = a_{i_1\dots i_{max}}\omega$

is a nonzero scalar multiple of ω, as required.

Note that for n even, this computation also shows that

$\Delta = \mathrm{C}\ell(W)\omega = (\Lambda^* W)\omega$.

as a vector space. In the last equality we again used that W is isotropic. In physics terms, this shows that Δ is built up like a Fock space by creating spinors using anti-commuting creation operators in W acting on a vacuum ω.

### Exterior algebra construction

The computations with the minimal ideal construction suggest that a spinor representation can also be defined directly using the exterior algebra Λ W = ⊕j Λj W of the isotropic subspace W. Let Δ = Λ W denote the exterior algebra of W considered as vector space only. This will be the spin representation, and its elements will be referred to as spinors.[27]

The action of the Clifford algebra on Δ is defined first by giving the action of an element of V on Δ, and then showing that this action respects the Clifford relation and so extends to a homomorphism of the full Clifford algebra into the endomorphism ring End(Δ) by the universal property of Clifford algebras. The details differ slightly according to whether the dimension of V is even or odd.

When dim(V) is even, V = WW where W′ is the chosen isotropic complement. Hence any vV decomposes uniquely as v = w + w with wW and w′ ∈ W. The action of v on a spinor is given by

$c(v) w_1 \wedge\cdots\wedge w_n = (\epsilon(w) + i(w'))\left(w_1 \wedge\cdots\wedge w_n\right)$

where i(w′) is interior product with w′ using the non degenerate quadratic form to identify V with V, and ε(w) denotes the exterior product. It may be verified that

c(u)c(v) + c(v)c(u) = 2 g(u,v),

and so c respects the Clifford relations and extends to a homomorphism from the Clifford algebra to End(Δ).

The spin representation Δ further decomposes into a pair of irreducible complex representations of the Spin group[28] (the half-spin representations, or Weyl spinors) via

$\Delta_+ = \Lambda^{even} W,\, \Delta_- = \Lambda^{odd} W$.

When dim(V) is odd, V = WUW, where U is spanned by a unit vector u orthogonal to W. The Clifford action c is defined as before on WW, while the Clifford action of (multiples of) u is defined by

$c(u) \alpha = \left\{\begin{matrix} \alpha&\hbox{if } \alpha\in \Lambda^{even} W\\ -\alpha&\hbox{if } \alpha\in \Lambda^{odd} W \end{matrix}\right.$

As before, one verifies that c respects the Clifford relations, and so induces a homomorphism.

### Hermitian vector spaces and spinors

If the vector space V has extra structure that provides a decomposition of its complexification into two maximal isotropic subspaces, then the definition of spinors (by either method) becomes natural.

The main example is the case that the real vector space V is a hermitian vector space (V, h), i.e., V is equipped with a complex structure J that is an orthogonal transformation with respect to the inner product g on V. Then V ⊗RC splits in the ±i eigenspaces of J. These eigenspaces are isotropic for the complexification of g and can be identified with the complex vector space (V, J) and its complex conjugate (V, −J). Therefore for a hermitian vector space (V, h) the vector space Λ
C
V (as well as its complex conjugate Λ
C
V) is a spinor space for the underlying real euclidean vector space.

With the Clifford action as above but with contraction using the hermitian form, this construction gives a spinor space at every point of an almost Hermitian manifold and is the reason why every almost complex manifold (in particular every symplectic manifold) has a Spinc structure. Likewise, every complex vector bundle on a manifold carries a Spinc structure.[29]

## Clebsch–Gordan decomposition

A number of Clebsch–Gordan decompositions are possible on the tensor product of one spin representation with another.[30] These decompositions express the tensor product in terms of the alternating representations of the orthogonal group.

For the real or complex case, the alternating representations are

• Γr = ΛrV, the representation of the orthogonal group on skew tensors of rank r.

In addition, for the real orthogonal groups, there are three characters (one-dimensional representations)

• σ+ : O(p, q) → {−1, +1} given by σ+(R) = −1, if R reverses the spatial orientation of V, +1, if R preserves the spatial orientation of V. (The spatial character.)
• σ : O(p, q) → {−1, +1} given by σ(R) = −1, if R reverses the temporal orientation of V, +1, if R preserves the temporal orientation of V. (The temporal character.)
• σ = σ+σ . (The orientation character.)

The Clebsch–Gordan decomposition allows one to define, among other things:

• An action of spinors on vectors.
• A Hermitian metric on the complex representations of the real spin groups.
• A Dirac operator on each spin representation.

### Even dimensions

If n = 2k is even, then the tensor product of Δ with the contragredient representation decomposes as

$\Delta\otimes\Delta^* \cong \bigoplus_{p=0}^n \Gamma_p \cong \bigoplus_{p=0}^{k-1} \left(\Gamma_p\oplus\sigma\Gamma_p\right)\, \oplus \Gamma_k$

which can be seen explicitly by considering (in the Explicit construction) the action of the Clifford algebra on decomposable elements αω ⊗ βω. The rightmost formulation follows from the transformation properties of the Hodge star operator. Note that on restriction to the even Clifford algebra, the paired summands ΓpσΓp are isomorphic, but under the full Clifford algebra they are not.

There is a natural identification of Δ with its contragredient representation via the conjugation in the Clifford algebra:

$(\alpha\omega)^*=\omega(\alpha^*).$

So Δ ⊗ Δ also decomposes in the above manner. Furthermore, under the even Clifford algebra, the half-spin representations decompose

$\begin{matrix} \Delta_+\otimes\Delta^*_+ \cong \Delta_-\otimes\Delta^*_- &\cong& \bigoplus_{p=0}^k \Gamma_{2p}\\ \Delta_+\otimes\Delta^*_- \cong \Delta_-\otimes\Delta^*_+ &\cong& \bigoplus_{p=0}^{k-1} \Gamma_{2p+1} \end{matrix}$

For the complex representations of the real Clifford algebras, the associated reality structure on the complex Clifford algebra descends to the space of spinors (via the explicit construction in terms of minimal ideals, for instance). In this way, we obtain the complex conjugate Δ of the representation Δ, and the following isomorphism is seen to hold:

$\bar{\Delta} \cong \sigma_-\Delta^*$

In particular, note that the representation Δ of the orthochronous spin group is a unitary representation. In general, there are Clebsch–Gordan decompositions

$\Delta\otimes\bar{\Delta} \cong \bigoplus_{p=0}^k\left(\sigma_-\Gamma_p\oplus \sigma_+\Gamma_p\right).$

In metric signature (p, q), the following isomorphisms hold for the conjugate half-spin representations

• If q is even, then $\bar{\Delta}_+ \cong \sigma_-\otimes \Delta_+^*$ and $\bar{\Delta}_- \cong \sigma_-\otimes \Delta_-^*.$
• If q is odd, then $\bar{\Delta}_+ \cong \sigma_-\otimes \Delta_-^*$ and $\bar{\Delta}_- \cong \sigma_-\otimes \Delta_+^*.$

Using these isomorphisms, one can deduce analogous decompositions for the tensor products of the half-spin representations Δ±Δ±.

### Odd dimensions

If n = 2k + 1 is odd, then

$\Delta\otimes\Delta^* \cong \bigoplus_{p=0}^k \Gamma_{2p}.$

In the real case, once again the isomorphism holds

$\bar{\Delta} \cong \sigma_-\Delta^*.$

Hence there is a Clebsch–Gordan decomposition (again using the Hodge star to dualize) given by

$\Delta\otimes\bar{\Delta} \cong \sigma_-\Gamma_0\oplus\sigma_+\Gamma_1\oplus\dots\oplus\sigma_\pm\Gamma_k$

### Consequences

There are many far-reaching consequences of the Clebsch–Gordan decompositions of the spinor spaces. The most fundamental of these pertain to Dirac's theory of the electron, among whose basic requirements are

• A manner of regarding the product of two spinors ϕψ as a scalar. In physical terms, a spinor should determine a probability amplitude for the quantum state.
• A manner of regarding the product ψϕ as a vector. This is an essential feature of Dirac's theory, which ties the spinor formalism to the geometry of physical space.
• A manner of regarding a spinor as acting upon a vector, by an expression such as ψvψ. In physical terms, this represents an electric current of Maxwell's electromagnetic theory, or more generally a probability current.

## Summary in low dimensions

• In 1 dimension (a trivial example), the single spinor representation is formally Majorana, a real 1-dimensional representation that does not transform.
• In 2 Euclidean dimensions, the left-handed and the right-handed Weyl spinor are 1-component complex representations, i.e. complex numbers that get multiplied by e±/2 under a rotation by angle φ.
• In 3 Euclidean dimensions, the single spinor representation is 2-dimensional and quaternionic. The existence of spinors in 3 dimensions follows from the isomorphism of the groups SU(2) ≅ Spin(3) that allows us to define the action of Spin(3) on a complex 2-component column (a spinor); the generators of SU(2) can be written as Pauli matrices.
• In 4 Euclidean dimensions, the corresponding isomorphism is Spin(4) ≅ SU(2) × SU(2). There are two inequivalent quaternionic 2-component Weyl spinors and each of them transforms under one of the SU(2) factors only.
• In 5 Euclidean dimensions, the relevant isomorphism is Spin(5) ≅ USp(4) ≅ Sp(2) that implies that the single spinor representation is 4-dimensional and quaternionic.
• In 6 Euclidean dimensions, the isomorphism Spin(6) ≅ SU(4) guarantees that there are two 4-dimensional complex Weyl representations that are complex conjugates of one another.
• In 7 Euclidean dimensions, the single spinor representation is 8-dimensional and real; no isomorphisms to a Lie algebra from another series (A or C) exist from this dimension on.
• In 8 Euclidean dimensions, there are two Weyl–Majorana real 8-dimensional representations that are related to the 8-dimensional real vector representation by a special property of Spin(8) called triality.
• In d + 8 dimensions, the number of distinct irreducible spinor representations and their reality (whether they are real, pseudoreal, or complex) mimics the structure in d dimensions, but their dimensions are 16 times larger; this allows one to understand all remaining cases. See Bott periodicity.
• In spacetimes with p spatial and q time-like directions, the dimensions viewed as dimensions over the complex numbers coincide with the case of the (p + q)-dimensional Euclidean space, but the reality projections mimic the structure in |p − q| Euclidean dimensions. For example, in 3 + 1 dimensions there are two non-equivalent Weyl complex (like in 2 dimensions) 2-component (like in 4 dimensions) spinors, which follows from the isomorphism SL(2, C) ≅ Spin(3,1).
Metric signature left-handed Weyl right-handed Weyl conjugacy Dirac left-handed Majorana–Weyl right-handed Majorana–Weyl Majorana
complex complex complex real real real
(2,0) 1 1 mutual 2 2
(1,1) 1 1 self 2 1 1 2
(3,0) 2
(2,1) 2 2
(4,0) 2 2 self 4
(3,1) 2 2 mutual 4 4
(5,0) 4
(4,1) 4
(6,0) 4 4 mutual 8 8
(5,1) 4 4 self 8
(7,0) 8 8
(6,1) 8
(8,0) 8 8 self 16 8 8 16
(7,1) 8 8 mutual 16 16
(9,0) 16 16
(8,1) 16 16

## Notes

1. ^ Spinors in three dimensions are points of a line bundle over a conic in the projective plane. In this picture, which is associated to spinors of a three-dimensional pseudo-Euclidean space of signature (1,2), the conic is an ordinary real conic (here the circle), the line bundle is the Möbius bundle, and the spin group is SL2(R). In Euclidean signature, the projective plane, conic and line bundle are over the complex instead, and this picture is just a real slice.
2. ^ Spinors can always be defined over the complex numbers. However, in some signatures there exist real spinors. Details can be found in spin representation.
3. ^ A formal definition of spinors at this level is that the space of spinors is a linear representation of the Lie algebra of infinitesimal rotations of a certain kind.
4. ^ More precisely, it is the fermions of spin-1/2 that are described by spinors, which is true both in the relativistic and non-relativistic theory. The wavefunction of the non-relativistic electron has values in 2 component spinors transforming under three-dimensional infinitesimal rotations. The relativistic Dirac equation for the electron is an equation for 4 component spinors transforming under infinitesimal Lorentz transformations for which a substantially similar theory of spinors exists.
5. ^ Formally, the spin group is the group of relative homotopy classes with fixed endpoints in the rotation group.
6. ^ More formally, the space of spinors can be defined as an (irreducible) representation of the spin group that does not factor through a representation of the rotation group (in general, the connected component of the identity of the orthogonal group).
7. ^ Geometric algebra is a name for the Clifford algebra in an applied setting.
8. ^ the Pauli matrices correspond to angular momenta operators about the three coordinate axes. This makes them slightly a-typical gamma matrices because in addition to their anti commutation relation they also satisfy commutation relations
9. ^ The metric signature relevant as well if we are concerned with real spinors. See spin representation.
10. ^ Whether the representation decomposes depends on whether they are regarded as representations of the spin group (or its Lie algebra), in which case it decomposes in even but not odd dimensions, or the Clifford algebra when it is the other way around. Other structures than this decomposition can also exist; precise criteria are covered at spin representation and Clifford algebra.
11. ^ This is the set of 2×2 complex traceless hermitian matrices.
12. ^ Except for a kernel[disambiguation needed] of $\{\pm 1\}$ corresponding to the two different elements of the spin group that go to the same rotation.
13. ^ Although there are several more intrinsic constructions, the spin representations are not functorial in the quadratic form, so they cannot be built up naturally within the tensor algebra.
14. ^ So the ambiguity in identifying the spinors themselves persists from the point of view of the group theory, and still depends on choices.
15. ^ The Clifford algebra can be given an even/odd grading from the parity of the degree in the gammas, and the spin group and its Lie algebra both lie in the even part. Whether here by "representation" we mean representations of the spin group or the Clifford algebra will affect the determination of their reducibility. Other structures than this splitting can also exist; precise criteria are covered at spin representation and Clifford algebra.

## References

1. ^
2. ^ a b Quote from Elie Cartan: The Theory of Spinors, Hermann, Paris, 1966, first sentence of the Introduction section of the beginning of the book (before the page numbers start): "Spinors were first used under that name, by physicists, in the field of Quantum Mechanics. In their most general form, spinors were discovered in 1913 by the author of this work, in his investigations on the linear representations of simple groups*; they provide a linear representation of the group of rotations in a space with any number $n$ of dimensions, each spinor having $2^\nu$ components where $n = 2\nu+1$ or $2\nu$." The star (*) refers to Cartan 1913.
3. ^ Named after William Kingdon Clifford,
4. ^ Named after Paul Dirac.
5. ^ Named after Hermann Weyl.
6. ^ Named after Ettore Majorana.
7. ^ Matthew R. Francis, Arthur Kosowsky: The Construction of Spinors in Geometric Algebra, submitted 20 March 2004, version of 18 October 2004 arXiv:math-ph/0403040
8. ^ *Wilczek, Frank (2009). "Majorana returns". Nature Phys. (Macmillan Publishers) 5 (9): 614–618. doi:10.1038/nphys1380. ISSN 1745-2473. (subscription required (help)).
9. ^ Xu, Yang-Su et al. (2015). "Discovery of a Weyl Fermion semimetal and topological Fermi arcs". Science Magazine (AAAS). doi:10.1126/science.aaa9297. ISSN 0036-8075. (subscription required (help)).
10. ^ Jean Hladik: Spinors in Physics, translated by J. M. Cole, Springer 1999, ISBN 978-0-387-98647-0, p. 3
11. ^ Graham Farmelo: The Strangest Man. The Hidden Life of Paul Dirac, Quantum Genius, Faber & Faber, 2009, ISBN 978-0-571-22286-5, p. 430
12. ^ Cartan 1913
13. ^ Tomonaga 1998, p. 129
14. ^
15. ^
16. ^ G. Juvet: Opérateurs de Dirac et équations de Maxwell, Commentarii Mathematici Helvelvetici, 2 (1930), pp. 225–235, doi:10.1007/BF01214461 (abstract in French language)
17. ^ F. Sauter: Lösung der Diracschen Gleichungen ohne Spezialisierung der Diracschen Operatoren, Zeitschrift für Physik, Volume 63, Numbers 11–12, 803–814, doi:10.1007/BF01339277 (abstract in German language)
18. ^ a b Pertti Lounesto: Crumeyrolle's bivectors and spinors, pp. 137–166, In: Rafał Abłamowicz, Pertti Lounesto (eds.): Clifford algebras and spinor structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919–1992), ISBN 0-7923-3366-7, 1995, p. 151
19. ^ The matrices of dimension N × N in which only the elements of the left column are non-zero form a left ideal in the N × N matrix algebra Mat(N, C) – multiplying such a matrix M from the left with any N × N matrix A gives the result AM that is again an N × N matrix in which only the elements of the left column are non-zero. Moreover, it can be shown that it is a minimal left ideal. See also: Pertti Lounesto: Clifford algebras and spinors, London Mathematical Society Lecture Notes Series 286, Cambridge University Press, Second Edition 2001, DOI 978-0-521-00551-7, p. 52
20. ^ a b Pertti Lounesto: Clifford algebras and spinors, London Mathematical Society Lecture Notes Series 286, Cambridge University Press, Second Edition 2001, DOI 978-0-521-00551-7, p. 148 f. and p. 327 f.
21. ^ D. Hestenes: Space–Time Algebra, Gordon and Breach, New York, 1966, 1987, 1992
22. ^ D. Hestenes: Real spinor fields, J. Math. Phys. 8 (1967), pp. 798–808
23. ^ These are the right-handed Weyl spinors in two dimensions. For the left-handed Weyl spinors, the representation is via γ(ϕ) = γϕ. The Majorana spinors are the common underlying real representation for the Weyl representations.
24. ^ Since, for a skew field, the kernel of the representation must be trivial. So inequivalent representations can only arise via an automorphism of the skew-field. In this case, there are a pair of equivalent representations: γ(ϕ) = γϕ, and its quaternionic conjugate γ(ϕ) = ϕγ.
25. ^ The complex spinors are obtained as the representations of the tensor product HR C = Mat2(C). These are considered in more detail in spinors in three dimensions.
26. ^ This construction is due to Cartan. The treatment here is based on Chevalley (1954).
27. ^ One source for this subsection is Fulton & Harris (1991).
28. ^ Via the even-graded Clifford algebra.
29. ^ Lawson & Michelsohn 1989, Appendix D.
30. ^


text:
# §2.8 Differential Equations with a Parameter

## §2.8(i) Classification of Cases

Many special functions satisfy an equation of the form

in which is a real or complex parameter, and asymptotic solutions are needed for large that are uniform with respect to in a point set in or . For example, can be the order of a Bessel function or degree of an orthogonal polynomial. The form of the asymptotic expansion depends on the nature of the transition points in , that is, points at which has a zero or singularity. Zeros of are also called turning points.

There are three main cases. In Case I there are no transition points in and is analytic. In Case II has a simple zero at and is analytic at . In Case III has a simple pole at and is analytic at .

The same approach is used in all three cases. First we apply the Liouville transformation1.13(iv)) to (2.8.1). This introduces new variables and , related by

2.8.2

dots denoting differentiations with respect to . Then

where

The transformation is now specialized in such a way that: (a) and are analytic functions of each other at the transition point (if any); (b) the approximating differential equation obtained by neglecting (or part of ) has solutions that are functions of a single variable. The actual choices are as follows:

2.8.5

for Case I,

2.8.6

for Case II,

2.8.7

for Case III.

The transformed equation has the form

with (Case I), (Case II), (Case III). In Cases I and II the asymptotic solutions are in terms of the functions that satisfy (2.8.8) with . These are elementary functions in Case I, and Airy functions (§9.2) in Case II. In Case III the approximating equation is

where as . Solutions are Bessel functions, or modified Bessel functions, of order (§§10.2, 10.25).

For another approach to these problems based on convergent inverse factorial series expansions see Dunster et al. (1993) and Dunster (2001a, 2004).

## §2.8(ii) Case I: No Transition Points

The transformed differential equation is

in which ranges over a bounded or unbounded interval or domain , and is or analytic on . The parameter is assumed to be real and positive. Corresponding to each positive integer there are solutions , , that depend on arbitrarily chosen reference points , are or analytic on , and as

with and

(the constants of integration being arbitrary). The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to . The regions of validity comprise those points that can be joined to in by a path along which is nondecreasing or nonincreasing as passes from to . In addition, and must be bounded on .

For error bounds, extensions to pure imaginary or complex , an extension to inhomogeneous differential equations, and examples, see Olver (1997b, Chapter 10). This reference also supplies sufficient conditions to ensure that the solutions and having the properties (2.8.11) and (2.8.12) are independent of .

## §2.8(iii) Case II: Simple Turning Point

The transformed differential equation is

and for simplicity is assumed to range over a finite or infinite interval with , . Again, and is on . Corresponding to each positive integer there are solutions , , that are on , and as

when . For and see §9.2. The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to . These results are valid when and are finite.

An alternative way of representing the error terms in (2.8.15) and (2.8.16) is as follows. Let be the real root of the equation

2.8.19

of smallest absolute value, and define the envelopes of and by

2.8.20,
2.8.21
,.

These envelopes are continuous functions of , and as

uniformly with respect to .

For error bounds, more delicate error estimates, extensions to complex and , zeros, connection formulas, extensions to inhomogeneous equations, and examples, see Olver (1997b, Chapters 11, 13), Olver (1964b), Reid (1974a, b), Boyd (1987), and Baldwin (1991).

For other examples of uniform asymptotic approximations and expansions of special functions in terms of Airy functions see especially §10.20 and §§12.10(vii), 12.10(viii); also §§12.14(ix), 13.20(v), 13.21(iii), 13.21(iv), 15.12(iii), 18.15(iv), 30.9(i), 30.9(ii), 32.11(ii), 32.11(iii), 33.12(i), 33.12(ii), 33.20(iv), 36.12(ii), 36.13.

## §2.8(iv) Case III: Simple Pole

The transformed equation (2.8.8) is renormalized as

We again assume with , . Also, is on , and . The constant () is real and nonnegative.

There are two cases: and . In the former, corresponding to any positive integer there are solutions , , that are on , and as

If , then there are solutions , , that are on , and as

Here ,

, and (2.8.28) again applies. For and see §10.2(ii). The expansions (2.8.29) and (2.8.30) are both uniform and differentiable with respect to . These results are valid when and are finite.

uniformly with respect to .

For error bounds, more delicate error estimates, extensions to complex , , and , zeros, and examples see Olver (1997b, Chapter 12), Boyd (1990a), and Dunster (1990a).

For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv).

## §2.8(v) Multiple and Fractional Turning Points

The approach used in preceding subsections for equation (2.8.1) also succeeds when is a multiple or fractional turning point. For the former has a zero of multiplicity and is analytic. For the latter and are both analytic at , () being a real constant. In both cases uniform asymptotic approximations are obtained in terms of Bessel functions of order . More generally, can have a simple or double pole at . (In the case of the double pole the order of the approximating Bessel functions is fixed but no longer .) However, in all cases with and or , only uniform asymptotic approximations are available, not uniform asymptotic expansions. For results, including error bounds, see Olver (1977c).

For connection formulas for Liouville–Green approximations across these transition points see Olver (1977b, a, 1978).

## §2.8(vi) Coalescing Transition Points

Corresponding to the problems for integrals outlined in §§2.3(v), 2.4(v), and 2.4(vi), there are analogous problems for differential equations.

For two coalescing turning points see Olver (1975a, 1976) and Dunster (1996a); in this case the uniform approximants are parabolic cylinder functions. (For envelope functions for parabolic cylinder functions see §14.15(v)).

For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order.

For a coalescing turning point and simple pole see Nestor (1984) and Dunster (1994b); in this case the uniform approximants are Whittaker functions (§13.14(i)) with a fixed value of the second parameter.

For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24.

For further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§13.21(ii), 14.15(ii), 14.15(iv), 14.20(viii), 30.9(i), 30.9(ii).

For examples of uniform asymptotic approximations in terms of Whittaker functions with fixed second parameter see §18.15(i) and §28.8(iv).

Lastly, for an example of a fourth-order differential equation, see Wong and Zhang (2007).


text:
# Sequences of groups, exact not just in etale but also in the Zariski topology

Given $A=k[[x,y]]$, with say $k=\mathbb{C}$ and $X=Spec(A)$. $B$ denotes a free $A$-algebra of rank $e^2$, actually we have $B=A[u,v]$ with $u^e=x$, $v^e=y$ and $uv=\xi_e vu$, where $\xi_e$ is an e-th root of unity. Now we have $M_n(B)$ and the subalgebra $D$, where entries under the diagonal are in $uB$ and otherwise in $B$.

This gives us an exact sequence of noncommutative groups, where $i: \lbrace x=0 \rbrace \hookrightarrow X$:

$1\rightarrow D^{\times} \rightarrow Gl_n(B) \rightarrow i_{\*}F \rightarrow 1$.

Why is this an exact sequence of sheaves in the Zariski topology on $X$? That is: why can i look at the induced exact sequence in Zariski cohomology?

This is stated $\href{http://www.math.lsa.umich.edu/courses/711/ordpages60-85.ps}{here}$ on page 3, around the $3\times 3$-diagram. Or do we need the whole diagram to see this?

I mean even in the simple example $1\rightarrow \mu_n \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 1$, where $\mathbb{G}_m \rightarrow \mathbb{G}_m$ is $x\mapsto x^n$, this sequence is not exact when we use the Zariski topology.

If it is easier, one can replace $B$ with $A$ and $uB$ with $xA$. The goal is to see that $H^1(U,D^{\times})=0$, i.e. every fractional reflexive left D-ideal is free. Here $U=X\backslash \lbrace \mathfrak{m} \rbrace$. May be there is an easier way to see this?

-
$D^\times$-torsors over some $X$-scheme $Y$ correspond to locally free (formally local in the flat topology) $D_Y$-modules of rank $1$. Such modules are locally free already in the Zeriski topology. Now, the obstruction to lifting a section of $i_\ast F$ is such a torsor and hence the section is liftable Zariski-locally. –  Torsten Ekedahl Jul 7 '11 at 16:57
I'm not so well-read in the area of torsors: Why is the obstruction to lifting a section Zariski-locally such a torsor (and in what topology)? And why correspond such torsors to locally free modules of rank 1? Since X is local and the modules are locally free, there is in fact only the trivial $D$-module, so there is no obstruction. Or is this fact not needed? Maybe you know a good book or some literature, where i can just read and learn that by myself. –  TonyS Jul 7 '11 at 17:59

Let me elaborate on Torstens comment.

The map $GL_n(B) \to i_*F$ in your example is an fppf $D^\times$-torsor over $i_*F$, a fact which can be checked easily directly from the definition of torsor which you can find in any of the referenced texts. A priori it is a torsor for the fppf topology, but it can be shown that any $D^\times$-torsor for the fppf topology is also a torsor for the Zariski-topology.

There are only a few groups with this property. They were studied by Serre and Grothendieck [GS58] who classified all such groups over algebraically closed fields. They called them "special groups" (probably because of the joy of saying things like: "the general linear group is special" or better still: "the special orthogonal group is not special"). If I recall correctly, the complete list is $SL_n$, $Sp_{2n}$, all connected affine solvable groups and extensions thereof (in particular, $GL_n$ is special). Your group is defined over a more general base, so the result doesn't apply directly. However, specialness for $D^\times$ can be proven with exactly the same methods as for instance for $GL_n$.

This seems to be a folklore result, and I don't know of any reference for a proof (although it is mentioned for instance in [Joy07, Definition 2.1]), but the idea is as follows. The key is to use flat descent for quasi-coherent sheaves. The group $D^\times$ is the automorphism group for $D$ viewed as a module over itself. Given any module $M$ which is fppf-locally isomorphic to $D$, which is the same as locally free of rank 1 by flat descent, you get a corresponding $D^\times$-torsor by taking the sheaf of $D$-module isomorphisms $Isom(M, D)$. Note that this sheaf has a natural action on the right by $Aut(D) \simeq D^\times$. This establishes a one-to-one correspondance between isomorphism classes of locally free $D$-modules of rank 1 and isomorphism classes of torsors [Gir70 III 2.5]. Since any locally free $D$-module allows a Zariski-trivialisation, so does any $D^\times$-torsor.

[GS58] Séminaire C. Chevalley; 2e année: 1958. Anneaux de Chow et applications

[Gir70] J. Giraud, Cohomologie non abélienne

[Vis05] Fantechi et al. FGA explained, Part 1

[Mil80] Milne, Étale cohomology

[Joy07] D. Joyce, Motivic invariants of Artin stacks and 'stack functions'. http://xxx.lanl.gov/abs/math.AG/0509722

-
Oh, thanks a lot. That is very helpful. Just to clarify: The map $Gl_n(B) \rightarrow i_{\*}F$ is a torsor in the Zarski-topology, that's why the sequence i'm interested in is exact in the Zariksi-topology? Is was confused because of the word obstruction. I thought this word implied that some group has to vanish for this to be true. –  TonyS Jul 8 '11 at 10:48
Yes it is. And by definition this just means that it is a pseudo-torsor which is an epimorphism as a morphism of sheaves in the Zariski-topology. That is, it has a section Zariski locally. –  Daniel Bergh Jul 8 '11 at 15:04
Okay. Thanks for your help. Now i have enough to read for the weekend :-). –  TonyS Jul 8 '11 at 16:06


text:
# Include section number when referencing enum item

I have something like this:

\section{a section name}
\subsection{a subsection name}
\begin{enumerate}
\item\label{itm:first} Item one
\item Item two
\item Item three
\end{enumerate}
I am saying something else about \ref{itm:first}

The bottom line of the output looks like this: I am saying something else about 1. Is there a way to include the section number the list item is in, in the reference to it? This would look like I am saying something else about 1.1.1

If I understand right you want normal numbering of the enumeration items but numbering with subsection numbering in the reference. There are two commands you can play with here, \theenumi and \labelenumi. They are normally (like in article) defined like:

\newcommand\theenumi{\arabic{enumi}} %% Format for references
\newcommand\labelenumi{\theenumi.}.  %% Printed by \item (add '.')

To get the special format for the references you can do

\renewcommand\labelenumi{\arabic{enumi}.}
\renewcommand\theenumi{\thesubsection.\arabic{enumi}}%% references with subsection numbering

But even if it is possible, I doubt it is recommended. The references is there for the reader to find a specific place in the text, and using different formats in the \item and the reference will be confusing.

An alternative way is to add the extra numbering to both the \item and the reference with

\renewcommand\theenumi{\thesubsection.\arabic{enumi}}

Then, with the normal definition of \labelenumi the same is used with an additional period.

A third alternative, that I would prefer, is to just in the text a specification of which enumeration list you mean, e.g. in Subsection~\ref{sec:Mysubsection}.

All the above versions are given in one file below.

\documentclass{article}
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand\labelenumi{\arabic{enumi}.}
\renewcommand\theenumi{\thesubsection.\arabic{enumi}}%% references with subsection numbering
\section{a section name}
\subsection{a subsection name}
\begin{enumerate}
\item\label{itm:first} Item one
\item Item two
\item Item three
\end{enumerate}
I am saying something else about item \ref{itm:first}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand\theenumi{\arabic{enumi}}%% Back to normal
\renewcommand\labelenumi{\theenumi.}%% Back to normal
\renewcommand\theenumi{\thesubsection.\arabic{enumi}}%% Add subsection number to item
\section{a section name}
\subsection{a subsection name}
\begin{enumerate}
\item Item one
\item \label{itm:second}Item two
\item Item three
\end{enumerate}
I am saying something else about item \ref{itm:second}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand\theenumi{\arabic{enumi}}%% Back to normal
\section{a section name}
\label{sec:Mysection}
\subsection{a subsection name}
\label{sec:Mysubsection}
\begin{enumerate}
\item Item one
\item Item two
\item \label{itm:third}Item three
\end{enumerate}
I am saying something else about item \ref{itm:third} in Subsection~\ref{sec:Mysubsection}.
\end{document}


text:
# About congruence of two triangles

In Euclidean geometry, we know that SSS, SAS, AAS (or equivalently ASA) and RHS are the only 4 tools for proving the congruence of two triangles. I am wondering if the following can be added to that list:-

If ⊿ABC~⊿PQR and the area of ⊿ABC = that of ⊿PQR, then the two triangles are congruent.

-

If two triangles (in fact, any two polygons) are similar, with scale factor $k$, then the ratio of their areas is $k^2$. If the areas are equal then $k^2=1$ which implies that $k=1$, and therefore the triangles are congruent.


text:
Formatted question description: https://leetcode.ca/all/1750.html

# 1750. Minimum Length of String After Deleting Similar Ends

Medium

## Description

Given a string s consisting only of characters 'a', 'b', and 'c'. You are asked to apply the following algorithm on the string any number of times:

1. Pick a non-empty prefix from the string s where all the characters in the prefix are equal.
2. Pick a non-empty suffix from the string s where all the characters in this suffix are equal.
3. The prefix and the suffix should not intersect at any index.
4. The characters from the prefix and suffix must be the same.
5. Delete both the prefix and the suffix.

Return the minimum length of s after performing the above operation any number of times (possibly zero times).

Example 1:

Input: s = “ca”

Output: 2

Explanation: You can’t remove any characters, so the string stays as is.

Example 2:

Input: s = “cabaabac”

Output: 0

Explanation: An optimal sequence of operations is:

• Take prefix = “c” and suffix = “c” and remove them, s = “abaaba”.
• Take prefix = “a” and suffix = “a” and remove them, s = “baab”.
• Take prefix = “b” and suffix = “b” and remove them, s = “aa”.
• Take prefix = “a” and suffix = “a” and remove them, s = “”.

Example 3:

Input: s = “aabccabba”

Output: 3

Explanation: An optimal sequence of operations is:

• Take prefix = “aa” and suffix = “a” and remove them, s = “bccabb”.
• Take prefix = “b” and suffix = “bb” and remove them, s = “cca”.

Constraints:

• 1 <= s.length <= 10^5
• s only consists of characters 'a', 'b', and 'c'.

## Solution

Use two pointers to point to letters at two ends of string s. If the letters at the two pointers are the same, then move the two pointers towards the middle of the string s, until a different letter is met.

If at one point, all the letters between the two pointers (including the letters at the two pointers) are the same, return 0. Otherwise, the pointers will stop moving once they point to different letters, and return the number of letters between the two pointers (including the letters at the two pointers).

class Solution {
public int minimumLength(String s) {
int left = 0, right = s.length() - 1;
while (left < right) {
char leftChar = s.charAt(left), rightChar = s.charAt(right);
if (leftChar != rightChar)
break;
while (left < right && s.charAt(left) == leftChar)
left++;
if (left == right)
return 0;
while (left < right && s.charAt(right) == rightChar)
right--;
}
return right - left + 1;
}
}