# Representations Are Everywhere

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1 Representations Are Everywhere Nanghua Xi Member of Chinese Academy of Sciences 1 What is Representation theory Representation is reappearance of some properties or structures of one object on another. I.M.Gelfand has said: All of Mathematics is some kinds of representation theory. The accurate mean is to make the algebraic structure of an object to reappear on an concrete object which is made up of linear transformations (or matrices). The algebraic structures which the representation theory concerns include: group, algebra, Lie algebra and so on. Algebraic structure can be determined by operation. Representation is one kind of homomorphism. Generally, two mathematical objects are related by map. The map preserving the operation represent the relations between structures, and it is called homomorphism. A representation is just a homomorphism, whose image is composed of linear transformations. The representation theory contains three parts: the representation theory of groups, the representation theory of algebras, the representation theory of Lie algebras. Representation of groups: homomorphism of group Group {invertible linear transformation on a linear space } Representation of algebras: homomorphism of algebra Algebra {linear transformation on a linear space} Representation of Lie algebras: homomorphism of Lie algebra Lie algebra {linear transformation on a linear space} A finite dimensional representation means the dimensional linear space is finite. 1

2 Another form of the finite dimensional representation: Representation of groups: homomorphism of group Group { n n invertible matrices on some field} Representation of algebras: homomorphism of algebra Algebra { n n matrices on some field} Representation of Lie algebra: homomorphism of Lie algebra Lie algebras { n n matrices on some field} The one dimensional representation of groups: Group homomorphism of group {Invertible elements on some field} This is also called the character of a group. Example GL n (F ) F, A det A. Here F is a field, F = F {0}, GL n (F ) = {n n invertible matrices on F }. ( ) x The Legendre symbol in quadratic reciprocity law is a character p of a cyclic group with degree n Gaussian sum in number theory: G(χ, σ) = χ(t)σ(t), t F p, where χ and σ are characters of additive groups Z/pZ = F p and multiplicative group F p respectively. The periodic functions on real number field are intrinsically functions on the unit cycle S = {e ix x R}. The quadratic integrable functions on S, which is denoted by H, is a Hilbert space. The characters of S are e inx, n Z, which are just the standard orthogonal basis of the space H. The language of module: the three parts of representation algebraic structure A homomorphism {some linear transformations on a linear space V} Example 2

3 V is a module of GL n (V ) (the set of invertible linear transformations). The polynomial ring F [x 1,..., x n ] is module of GL n (F ). The polynomial ring F [x 11, x 12,..., x n,n 1, x nn ] is a module of GL n (F ). The tangent space on the unit element of Lie group is the representation of this Lie group, and this representation is called adjoint representation. The cohomology group H i (X, L) is GL n (C). Where X is a flag manifold whose elements are the subspace filter of C n : V 1 V 2 V n = C n, dim V i = i, and L : is a holomorphic line bundle on X. L 2 (C) is a representation of SL 2 (C) L 2 (G/Γ) is a representation of G. G: Lie group; Γ: discrete subgroup with V ol(g/γ) being finite. The orbit wave functions of a single electron generate the representation of orthogonal group SO(3). The spin wave functions of a single electron generate the representation of unitarian group SU(2) By using the ten dimensional representation, Gell-Mann predicted the existence of Ω-particle, which was later proved by experiment. Quiver Γ: n 1 n Representation of Γ: {V i, f i 1 i n, 1 j n 1}, where V i are linear spaces over the field F, and f j are the linear transformation V j V j+1. 2 Basic idea of representation symmetry and linearization: the algebraic structure reflect symmetry; the representation of algebraic structures give their linearization, also reflect some symmetry of the associated linear space. Basic problems of representation theory 3

4 What kind of representation is the most basic? How can a general representation be constructed by the basic ones? How to construct the most basic representations? The properties of the most basic representations, such as classification, dimension, character and so on. The properties of some naturally constructed representations. Subrepresentation A: Algebraic structure (such as group, Lie algebra and so on) V : the representation of A Subrepresentation of V : A invariant subspace of V. Example: V = C n is naturally the representation of the symmetric group S n, and {(a, a,, a) a C} is a subrepresentation of V. Irreducible representation: A representation who has no subrepresentation except the zero representation and itself is called an irreducible representation. Irreducible representations are the most basic representations. Examples: one dimensional representation V is the irreducible representation of GL(V ) V is the irreducible representation of gl(v ) All of the complex polynomials with n variables and degree i form an irreducible representation of GL n (C) sl n (C) is the irreducible representation of GL n (C) gl n (C) is not the irreducible representation of GL n (C) V V is not the irreducible representation of gl(v ). L 2 (C) is the irreducible representation of SL 2 (C) Homomorphism and Isomorphism A homomorphism of representation (also called homomorphism of module) 4

5 is a linear map which preserves the action of representations. Let U, V be two representations of A, and φ : U V be a linear map such that φ(au) = aφ(u), a A. An isomorphism of representation is a invertible linear map which preserve the action of representation. Lemma 2.1 (Schur lemma) Every nonzero homomorphism between irreducible representations is isomorphism. Classification of irreducible modules A theorem on finite groups: if the representation space is a complex linear space, then The number of irreducible representations is equal to the number of the conjugate classes of the finite group; Every representation is a direct sum of irreducible representations; The quadratic sum of dimension of all irreducible representations is equal to degree of the group. Example We consider symmetric groups: The number of complex irreducible representations of the symmetric group S n is the partition number P (n) of n; The dimension of an irreducible representation can be calculated; If the character p of the base field F satisfies p n, then the classification of irreducible representations is known. There are several theorem on the classification of irreducible representation. Theorem 2.2 (Pontryagin duality) Let S be the unit circle, then we have following correspondences: {The irreducible unitarian representations of S} Z; {The irreducible unitarian representations of Z} S Theorem 2.3 (Peter-Weyl) For any compact Lie group G, 5

6 The subspace spanned by the matrix coefficients of finite dimensional unitarian representation is dense in L 2 (G). Every complex irreducible representation is a unitarian representation of finite dimension. Every unitarian representation is completely reducible. The multiplicity of every irreducible unitarian representation appearing in L 2 (G) is equal to the dimension of the representation. Theorem 2.4 (Weyl) Character formula. Theorem 2.5 For general linear groups, we have the following one to one correspondence: {The finite dimensional irreducible representations of GL n (C)} {(a 1, a 2,..., a n ) Z n a i a i+1 }. The characters of irreducible representations are given by Weyl s formula. Theorem 2.6 For general linear Lie algebras, we have the following one to one correspondence: {The finite dimensional irreducible representation of gl n (C)} {(a 1, a 2,..., a n ) Z n a i a i+1 }. The characters of irreducible representations are given by Weyl s formula. Theorem 2.7 For matrices algebra, let V be a n dimensional vector space on field F, then we have: The irreducible representation of End(V ) is only V itself; The irreducible representation of M n (F ) is only F n. Theorem 2.8 For the algebraic completion F p of Z/pZ, we have the following one to one correspondence: {The irreducible representation of GL n ( F p )} {(a 1, a 2,..., a n ) Z n a i a i+1 }. The character of the irreducible representation is still unclear. 6

7 Theorem 2.9 Every irreducible representation of gl n ( F p ) is finite dimensional. Theorem 2.10 For quiver Γ : n 1 n we have: The irreducible representations of Γ are: E i = (0,..., 0, F, 0,..., 0) where F is at the place of the ith component. Theorem 2.11 The classification of irreducible representation of SL 2 (C) is as follows: Trivial representation Unitarian major column: P k,iy, (k Z, y R) Complement column: C x, (0 < x < 2), in which the only isomorphism is P k,iy = P k, iy 3 Research method Algebra, Analysis, Differential geometry, Algebraic geometry, Topology Algebraic method Representation theory of finite groups: G: finite group; H: subgroup; F : field; F [G]: group algebra. Consider the properties of groups algebras and induced representation: M: representation of G Ind G HV = F [G] F [H] V. χ M : G F, g trace of linear transformation. Representation theory of Lie algebras and Lie groups: G: Lie group; g: Lie algebra; h: Lie subalgebra. Consider the general enveloping algebra U(g) and the induced representation: Ind g h V = U(g) U(h) V. 7

8 Representation theory of algebraic groups and quantum groups G: Algebraic group on algebraic closure field k; k[g]: coordinate algebra. Consider the representation of comodule and hyperalgebra of k[g]. The representation theory of quantum groups is parallel to the representation theory of Lie groups and Lie algebraic groups. Representation theory of topological groups and Lie groups. G: local compact group; X: local compact Hausdorff space. G acts on X continually and X has a G-invariant metric. The question is: how to compose L 2 (X) to irreducible presentation? Analytic method The representation theory of algebraic groups on p-adic fields: G: p-adic group, for example: GL n (Q p ); F : field, for example: complex field. Consider C (G, F ) = {localconstantg F }. This is a smooth representation. Differential geometric method The representation of Lie groups: G: Lie group; H: close subgroup. Consider the geometry on the homogeneous space H/G. Algebraic geometric method Representation theory of algebraic groups Representation theory of finite groups Representation theory of Lie algebras Example F : algebraic closed field, such as C, F p ; G: algebraic group on F, such as GL n (F ), SL n (F ), Sp 2n (F ), SO n (F ),... Research: H i (X, L), where X is algebraic variety which is acted by G, and L is the G equivariant vector bundle on X; F q : finite field with q elements. Finite Lie type groups: GL n (F q ), SL n (F q ), Sp 2n (F q ), SO n (F q ),... Finite Lie type groups are special subgroups of algebraic groups are the invariant groups of Frobenius map. Using l-adic cohomology to construct the representation of groups: Deligne- Lusztig theory. Representation theory of complex semisimple Lie algebras: Kazhdan- Lusztig conjecture chl W = y W( 1) l(w ) l(y) P y,w (l)chm y 8

9 This conjecture is proved by Beilinson-Bernstein and Brylinski-Kashiwara in Topological method Tools: fiber bundle, characteristic class, cohomology, K-theory and so on. Example X: flag manifold, G = GL n (C); N : the set of nilpotent linear transformations on C n ; Z = {(ξ, x 1, x 2 ) ξx i = x i } N X X, K G C (Z) is isomorphic to affine Hecke algebra, which played crucial role in the proof of the conjecture proposed by Deligne-Langands on affine Hecke algebra. 4 History character in number theory Gauss: Gauss sum, Disquisitiones Arithmeticae, 1801 Dirichlet: L function (1837), L(χ, s) = χ(n) n=1 n s representation of finite abelian groups Dederkind: definition of the character of some finite abelian groups, 1878 Webber: definition of the character of finite abelian groups, 1881; Lehrbuch der Algebra, 1896 representation of finite groups Dederkind: decomposition of groups determinant of finite abelian groups, 1880; G = {g 1, g 2,..., g n }, x gi : indefinite elements; the determinant x gi g j 1 is called the group determinant of G Frobenius: character theory, the orthogonal relations of the irreducible character, 1896; definition of representation, character of representation, 1897; induced representation, Frobenius reciprocal law, 1898; computation of character: P SL 2 (p)(1986), S n (1900), A n (1901) Burnside: application of finite dimensional representation theory G = p a q b G solvable, 1904; Conjecture: Finite group with odd degree is solvable,

10 Schur: Schur lemma, 1905; introduction of the projective representation, 1907: G GL n (C)/{scalar matrices} arithmetic properties of representation, Schur character, 1906 Noether: introduce group algebra F [G] and using the language of module, 1929, which is the begin of modern theory. Brauer: modular representation theory, the modular representation theory of k[g], chark G, classification of irreducible module, 1935; relation between modular representation theory and general representation theory, relation between decomposing matrices and Cartan matrices: C = D D, 1937; p-piece theory and the introduction of defect group Finite dimensional representation of Lie groups and Lie algebras For SL 2 (C) and sl 2 (C) S.Lie: determining the irreducible representations, 1893 E.Cartan, G.Fanp: complete reducibility of representation A.Hurwitz: unitarian technique, complete reducibility of the representation of SL n (C) and SO n (C), 1897 Casimir: introduction of Casimir operator, proved the complete reducibility of the representation of sl 2 (C), introduced general Casimir operator, in 1931 E.Cartan: construction of all irreducible representations of complex semisimple Lie algebras, 1913 Schur: the characters of SU n and SO n, complete reducibility, 1922 H.Weyl: proof of the complete reducibility of complex semisimple Lie groups, complex semisimple Lie algebras and compact Lie groups; the character of representation, 1926 Peter-Weyl: Peter-Weyl theorem, 1927; Gruppentheorie und Quantenmechanik Pontryajin, van Kampen: the Pontryajin duality of the compact commutative topological groups, van der Waerden: algebraic proof of the complete reducibility of finite dimensional representation of semisimple Lie algebras 10

11 Brauer: another algebraic proof of above result, 1936 Infinite dimensional representation of Lie groups and Lie algebras E.Winger: irreducible unitarian representation of Poincare groups, 1939 V.Bargmann: irreducible unitarian representation of Lorentz groups, 1947 I.M.Gelfand, M.A.Naimark: classification of irreducible unitarian representation of SL 2 (C), 1947 After 1950 Mathematicians in this branch of mathematics include: I.M.Gelfand, Harish-Chandra, Selberg, A.Weil, Grothendieck, Borel, Langlands, Deligne, Kazhdan, Drinfeld, Lafforgue, Bernstein, Beilinson, V.Kac, Lusztig,... 11

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