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AFFINE LIE ALGEBRAS, THE SYMMETRIC GROUPS, AND ALGEBRAIC GROUPS MALKA SCHAPS 1. Ordinary Representations of Groups For any finite groups G = {g1 , . . . , gs } and field F , a representation of G over F is a group homomorphism ρ : G → GLr (F ), and r is called its degree. Let F G be the corresponding group algebra, with F -vector space basis hg1 , . . . , gr i and algebra multiplication induced by the group multiplication law and linearity. Then the representation ρ induces an algebra homomorphism ρ̃ : F G → GLr (F ), P P where ai gi ∈ F G is mapped to ai ρ(gi ). The representation ρ can be recovered from ρ̃ by restricting to G, and we will consider them interchangeable. The most compact description of ρ or ρ̃ is given by specifying the images of a set of generators of G, as matrices satisfying the defining relations of the group. Two representations ρ and τ are considered equivalent if they differ only by a change of basis in the underlying vector space F r on which the matrices act. More formally, ρ τ if there exists an invertible r × r matrix P such that for each G ∈ G, tau(g) = P −1 ρ(g)P . The “atomic particles” of representation theory are the indecomposable representations, those which are not equivalent to a direct sum of representations. We assume, for the remainder of this section, that F satisfies the following two conditions, in which case we say that we are studying the ordinary representations of the finite group G. • (generic characteristic): F is a field of characteristic 0, or of positive characteristic p such that p - |G|. • (F sufficiently large): F contains a primitive root of unity ζ for |G| = s. If these two conditions hold, then the indecomposable representations are, in fact, irreducible, having no subrepresentations, and then by Maschke’s theorem, we can write F G as a direct sum of simple Date: Nov. 16-30, ’10. 1 2 MALKA SCHAPS algebras and decompose F G in the form FG ∼ = t M Mni (F ). i=1 Each component determines a unique irreducible representation ρi and these give a complete set of representatives of irreducible representations. The size ni of the matrix block is called the degree of the representation. The inverse images ei of the identity matrices Ini of the t components form a basis for the center Z(F G) of the group algebra F G. A representation ρ is completely determined by its character χρ , a map from G into F given by χρ (g) = tr(ρ(g)). Since the trace is invariant on conjugacy classes of matrices, χρ is a class function on G, that is, fixed on conjugacy classes. The character of the irreducible representation ρi will be denoted by χi for brevity. The addition of characters corresponds to direct sums of representations, and the componentwise product of characters corresponds to tensor products of representations. The conjugacy class sums form an alternative basis for the center Z(F G), and thus there are exactly t conjugacy classes. We will order the elements of G so that g1 is the identity element e of the group and gj , for j = 1, . . . , t are representatives of the distinct conjugacy P classes. The conjugacy class [gj ] will be denoted by Cj and its class sum g gj g bj . will be denoted by C We recall that we have assumed that F is of generic characteristic and is sufficiently large. The character table X is the t × t square matrix [χi (gj )]. For any i, j the character value χi (gj ) is the sum of the eigenvalues of ρi (gj ), and since gj is of finite order dividing the order s of G, the entries in the character table can be expressed as powers of ζ, making them more or less independent of the actual choice of the field F , as long as it satisfies our assumptions. If N is the diagonal matrix with the degrees along the diagonal, and M is the diagonal matrix with conjugacy class sizes mj = |Cj | along the diagonal, then W = N −1 XM , the matrix of central characters, gives a base change matrix between the two bases of Z(F G), in the sense that X Ĉj = wij ei . Since W, N and M are all invertible, so is X. 2. Ordinary representations of the symmetric groups For the symmetric groups Sd , the conjugacy classes consist of all elements with the same cycle structure. Each cycle structure can be 3 represented by a partition of d, and thus there is a natural one-toone correspondence between conjugacy classes and partitions. It is considerably more difficult, though standard, to show that the irreducible representations are also in one-to-one correspondence with the partitions of d, and we will assume this to be known. The character corresponding to a partition λ will the denoted by χλ , and the corresponding degree by nλ . The conjugacy class class representative with the integers {1, . . . , n} inserted in cycles of lengths λ1 , . . . , λk will be denoted by gλ . The representation corresponding to the partition with one part, (d), is the trivial representation of degree 1, sending every group element to 1, and the representation corresponding to the partition with d parts, (1, 1, . . . , 1) is the alternating representation, also of degree 1, which sends each group element to 1 if the permutation is even and to −1 if the permutation is odd. The partition is represented by its Young diagram. The partition is regarded as having been built up one node at a time from the empty partition, in such a way that each intermediate diagram is a partition. Each such path is recorded by placing the integers from 1 to n in the nodes of the diagram, creating a standard tableau in which the integer are increasing on every row and column. The degree nλ of the representation ρλ is the number of such paths and the set of standard tableaux may be taken as a basis. The defining property of this representation is that it is simultaneously induced from the trivial representation of the subgroup of Sd stabilizing the rows and from the alternating representation of the subgroup of Sd stabilizing the columns. We now leave the standard material which has been known in essence since the time of Frobenius 150 years ago, and describe a new approach to the representations of the symmetric group which goes back about 15 years. We retain the Young diagrams, but add a new label, a residue to each node. These new labels will be constant on diagonals, will increase by 1 to the right and will decrease by 1 as we go down. For simplicity, we will give the main diagonal the residue 0. Each Young diagram is uniquely described by the lengths of its diagonals, which are given by its content, a list of the number of nodes of each residue. A standard tableau is described by a list of the residues which are encountered as one builds up the Young diagram node by node. We now use this new labeling of nodes to connect the representations of the symmetric groups to the weights of an infinite dimensional Lie algebra A∞ . This Lie algebra is formed like the Kac-Moody algebras we have been studying, except that it has an infinite Cartan matrix, which looks like the classic Cartan matrix of type A extended infinitely in both directions, and an infinite Chevalley basis consisting of the union of sets {ei , fi , hi )} for i ∈ Z. There is also an infinite set of elements . . . , Λ−2 , Λ−1 , Λ0 , Λ1 , Λ2 , . . . , 4 MALKA SCHAPS which are dual to the hi as functionals, in the sense that hhi , Λj i = δij . We consider a highest weight module P for this Lie algebra, with highest weight Λ0 . The weight µ = Λ0 − i∈Z γi αi will be a weight of a homogeneous element of this module if and only if γ = (. . . , γ−1 , γ0 , γ1 , . . . ) is a sequence of positive integers with only a finite number of nonzero terms, and if, furthermore, it is the content of a partition of some nonnegative integer d. The element v ∅ is the unique basis element of weight Λ0 and is referred to by physicists as the vacuum vector. The basis elements with weight µ will be parameterized by sequences i which will determine the standard tableaux. The same sequence i determines the representation of the basis element v i as a product fid fid−1 . . . f1 v ∅ . The number of sequences producing the weight µ is exactly the degree of the corresponding representation of Sd . At any partition λ0 along the way, corresponding to a weight µ0 with content γ 0 , the addable nodes are those with residue i for which hµ0 , hi i > 0. The only possible values are −1, 0, 1 and the sum of all the values must be 1, so it is always possible to continue. Example 2.1. We give the residues for the partition (4, 2, 1). For compactness we use a, b, . . . to designate −1, −2, . . . . 0 1 2 3 a 0 b The standard tableau 1 2 3 4 5 6 7 corresponds to a sequence i = (0, 1, 2, 3, −1, 0, −2). The standard tableau 1 4 6 7 2 5 3 corresponds to a sequence i = (0, −1, −2, 1, 0, 2, 3). 3. Modular representations of the symmetric groups and Iwahori-Hecke algebras at root of unity We now turn to the representation theory of the symmetric group over a field whose characteristic e does divide d!, the so-called modular case. Dipper and James discovered that in this case the representation theory is similar to that of an Iwahori-Hecke algebra in which ξ is an e-th root of unity in F (which has characteristic different from e). The symmetric group case is called the degenerate case, and the remaining 5 values of ξ give non-degenerate Hecke algebras, but the two cases can be treated together with only minor modifications. We now define the Iwahori-Hecke algebra Hd (F, ξ). Let ξ be an element of F ∗ . We define an algebra Hd with generators T1 , . . . , Tn−1 over the field F , using relations (Ti + 1)(Ti − ξ) = 0, 1 ≤ i ≤ n − 1 Ti+1 Ti Ti+1 = Ti Ti+1 Ti 1 ≤ i ≤ n − 2 Ti Tj = Tj Ti , 1 ≤ i < j ≤ n − 1, |i − j| ≥ 2 If ξ is not a root of unity, then the representation theory of Hd is essentially the same as the ordinary representation theory of Sd , with the algebra being semi-simple and the irreducible representations parameterized by the partitions of d. If the field F is generic for Sn and ξ = 1, then we get the symmetric group. There is an analog of this construction for the so-called cyclotomic Hecke algebras HdΛ , in which the blocks correspond to weights with highest weight Λ but we will not go into that theory in these lectures. The main difference is in the meaning of the residues label the nodes of a partition. These are now taken to be elements of I = Z/eZ, which we will represent by numbers in the set {0, 1, . . . , e − 1}. In the degenerate case they represent elements of the base field of F , and in the nondegenerate case they represent powers of ξ. More formally, we define ( i (if ξ = 1) ν(i) ≡ ξ i (if ξ 6= 1). These elements ν(i) enter the theory as the eigenvalues of the following elements of Hd , known as Jucys-Murphy elements: ( Lr ≡ (1, r) + (2, r) + · · · + (r − 1, r) ξ 1−r Tr Tr−1 . . . T2 T1 T1 T2 . . . Tr−1 Tr (if ξ = 1) 1 ≤ r ≤ d. (if ξ = 6 1); Note that in the degenerate case, L1 = 0 and L2 = (1, 2). The Lr commute with each other and form a commutative subalgebra Z of Hd known as the Gelfand-Zetlin subalgebra. The restriction of any Hd module M to this subalgebra breaks up into a direct sum of linear representations, with eigenvalues of the form ν(i). The module induced from an irreducible module for Hd−1 will be a direct sum of indecomposable projective modules for Hd , each with irreducible socle and each socle with a single eigenvalue of the Jucys-Murphy element Ld . It is this choice of idempotents which distinguishes among various components in the induced modules. As before, the blocks correspond to weights in an affine Lie algebra, (1) but now the algebra is Ae−1 . This has a Dynkin diagram which is a 6 MALKA SCHAPS cycle of length e. The Cartan matrix is the Cartan matrix of Ae with an additional −1 at each end of the off diagonal. Thus for e = 5 it would be 2 −1 0 0 −1 −1 2 −1 0 0 0 −1 2 −1 0 0 0 −1 2 −1 −1 0 0 −1 2 In the modular case, it is no longer true that every indecomposable representation is irreducible. The blocks now correspond to collections of partitions with a common core, obtained by removing a fixed number w of rim hooks of length e. Each rim hook will contain a complete set of residues. In terms of the affine Lie algebra, removing rim hooks corresponds to adding copies of the null root. The irreducible modules correspond to those partitions which are erestricted, i.e., do not have e consecutive columns of the same length. (This is dual to the more traditional notion of e-regular, under the duality which flips the partitions across the main diagonal.) There is a diagram called the crystal graph, due originally to Kashiwara, which describes the way that the irreducible representations are built up by induction. Example 3.1. Let us consider the Young tableau in Example 2.1 for e = 3. 0 1 2 0 2 0 1 In this case there are three simple roots α0 , α1 , α2 to the affine Lie algebra, and the null root is δ = α0 + α1 + α2 .The corresponding weight in the affine Lie algebra is Λ0 − 3α0 − 2α1 − 2α2 . We can remove two rim hooks, which corresponds to adding two copies of δ, giving Λ0 − α0 . −−−− 2 0 1 ,− 1 2 0, 0. 7 We will now give that part of the crystal graph containing all the paths leading to (4, 2, 1). ∅ ↓ 0 . & 0 2 0 1 & . 0 1 2 2 ↓ . 0 2 0 1 1, 2 ↓ 0 1 2 1 0 1 2 0 .& .↓ 0 1 2 2 1 0 1 2 2 0 ↓ & & ↓ 0 1 2 0 1 . 0 1 2 2 0 1 0 1 2 0 2 0 & ↓ 0 1 2 0 2 0 1 Each i-string, for i ∈ I, is actually an sl2 -representation. As one goes down an i-string, the i-nodes are added starting at the bottom left and working along to the upper right. We give a few strings of this type. Note that the second 0-string is not contained in the diagram because it does not lead to the partition (4, 2, 1). 0 0 1 0 2 2 i=1: 2 → 1 → 1 . 8 MALKA SCHAPS 0 1 0 1 2 0 1 2 i=2: → → 2 . 0 1 0 1 2 i=0: → 2 0. 0 0 1 1 2 2 → 0. i=0: We recall that the finite Lie algebra corresponding to an affine Lie algebra is obtained by removing the simple root α0 , the corresponding Chevalley generators, and the first row and column of the Cartan matrix. In terms of the crystal graph, we define a floor to be the set of all vertices with a fixed integral coefficient for α0 . This will then correspond to a homogeneous basis for a representation of the finite Lie algebra, which in our case is sle . The irreducible representations of sle are highest weight representations, with highest weight λ1 Λ1 + · · · + λe−1 Λe−1 , for some partition λ of an integer d. The length of an i-string going out from the highest weight vector will be λi . Most of the floors will correspond to reducible representations of sle , but even these will be highly symmetrical because the finite Weyl group will act on each floor. For sl3 , the irreducible representations can be diagramed as irregular hexagons with alternating sides of integral lengths a and b. The finite Weyl group is S3 , where the axes of the reflections are the lines connecting the midpoints of opposite sides, and the 3-cycles permute the sides of equal length. If b = 0, the hexagon reduces to a triangle. One can find such a hexagon in the diagram above by considering all partitions with 0-content equal to 1. There is also a reduced version of the crystal graph in which the vertices correspond to blocks of the symmetric groups instead of to irreducible representations. Let P (Λ) be the set of weights of occurring in a highest weight module with highest weight Λ. The reduced crystal graph has a geometric representation in which the vertices are the integer lattice points in Re determined by the contents in P (Λ). Example 3.2. If we redo the diagram in Example 3.1 as a part of the reduced crystal graph, letting a block with core ν and weight w be represented by ν w , we get the following, where we have now included all blocks for d ≤ 7, including one which was not in the previous diagram. Note the symmetry between left and right, which is just taking the dual partition and multiplying the residues by −1. 9 ∅0 ↓ 0 0 . 0 2 & 0 & 0 1 0 0 1 2 1 0 . 1 . 0 1 2 2 0 0 ↓ 0 2 ∅ ↓ & 1 .& 0 1 2 2 1 1 ↓ 0 1 2 0 2 0 .& ↓ & 0 & ↓ . 1 ↓ 0 1 2 0 0 2 ∅2 ↓ 0 0 1 0 0 . 2 In this representation, one can see the effect of the shift by δ, which simply adds one to each exponent. One can also see that the exponents are symmetric along the strings, increasing toward the center. The weight space of an affine Lie algebra has two different bases. One is {Λ0 , α0 , . . . , αe−1 } and we have been representing our weights in this basis, using the components of the content as the coordinates of the αi . The second basis is obtained by taking the dual basis of the hi (or αi∨ ) and adding δ, giving {Λ0 , Λ1 , . . . , Λe−1 , δ}. The hub of a block is the set of coordinates of the weights of the affine Lie algebra with respect to the Λi , moding out by δ. The components in the hub represent the weights of the hi in the sl2 representation 10 MALKA SCHAPS along the i-string. We recall that an affine lie algebra has a suitably normalized central element c. The level of a weight µ is hµ, ci and in our case is just the sum of the coordinates of the hub. Since the level of any simple root is always zero, and thus in any highest weight representation, the level is fixed. In the symmetric group case, since the highest weight is Λ0 , which has level 1, the sum of the coordinates of any hub is 1. The i-th coordinate of the hub corresponding to any weight or content, gives the position of the corresponding vertex of the reduced crystal graph along its i string, where each application of ei increases the weight by 2 and each application of fi decreases the weight by 2. The set of all weights for which adding a copy of δ is no longer a weight in P (Λ0 ) is called the set of maximal weights, max(Λ0 ). In the chart below, the exponent on each hub is the number of copies of δ which can be added, and the elements of max(Λ0 ) are those vertices with exponent 0. (1, 0, 0)0 ↓ (−1, 1, 1)0 . & (0, 2, −1)0 (0, −1, 2)0 & . (1, 0, 0)1 . ↓ 0 & 1 (2, 1, −2) (−1, 1, 1) ↓ .& (0, 2, −1)1 ↓ & (−2, 3, 0)0 (2, −2, 1)0 .& ↓ (3, −1, −1)0 ↓ . (1, 0, 0)2 & ↓ (0, −1, 2)1 ↓ (−2, 0, 3)0 . (−1, 1, 1)2 4. Representations of algebraic groups We now come to the algebraic groups. We give a description in general of algebraic groups, before restricting to the finite groups among them. An algebraic group over a field L is a subgroup of GL(n, L) which can be described by polynomial equations {r1 (x), . . . , ru (x))}, with x = xi j ni,j=1 . Associated to the group is a coordinate ring, a commutative ring A = L[x, y]/(r1 , . . . , ru , det(x)y − 1). The most commonly studied groups are the general linear groups, orthogonal or unitary groups and 11 symplectic or spin groups, each category having its special (determinant 1) and projective (no scalars) versions. The field L can be R, C, a p-adic field, or a finite field Fq , for some prime power q, in which case the group is finite. The groups over R are called Lie groups and are studied with the tools of differential geometry. The groups over C are called complex Lie groups and are studied with the tools of algebraic geometry. Those over p-adic fields are studied with the tools of algebraic number theory, while the finite groups are generally the province of group theorists and combinatorists. The representation theory is simplest for the simple complex Lie groups SLn (C), SOn (C), and SP2 n(C), the so-called classical groups. In this case the representations are controlled by the corresponding Lie algebras sln (C), son (C), and sp2n (C). These representations are parameterized by partitions and are intimately connected to the representations of the symmetric group parameterized by the same partition in a way that we will now explain 4.1. The Weyl construction. The complex Lie group in GLn (C) has a natural representation V of dimension n, which is given by matrix multiplication on Cn on the left, where the vectors are written a columns. On the other hand, the tensor product V ⊗d has a natural action of Sd on the right by permutation of factors. V ⊗d Let F be any field generic for Sd , which surely includes C. We know that M F Sd ∼ Uλ , = where uλ ∼ = Mnλ (F ), and where nλ is the degree of the representation ρλ , which is given by the number of standard tableaux. As a left F Sd -module, Uλ is a direct sum of nλ copies of the irreducible module corresponding to the representation ρλ . For any vector space V over F , we have (1) V ⊗d = V ⊗d ⊗F Sd F Sd (2) = V ⊗d ⊗F Sd ⊕Uλ (3) = ⊕V ⊗d ⊗F Sd Uλ Each of the components of this direct sum is a GL(V ) modules, and decomposes into a direct sum of nλ isomorphic GL(V ) − Sd modules. In order to choose just one of these, we must go into more detail about the representation of Sd corresponding to a given partition λ, and, in fact, to a specific standard tableau T for λ. Let us define PT to be the set of elements of Sd which preserve the rows of T when operating on the numbers in the nodes of the Young diagram of λ defining T , and let Qλ be the subgroup of Sd preserving the columns of T . In the first tableau for (4, 2, 1) in Example 3.1, we would have PT = S{1,2,3,4} × S{4,5} × S{7} 12 MALKA SCHAPS and QT = S{1,5,7} × S{2,6} × S{3} × S{4} . We now define two elements of Sd associated to the tableau T , X aT = σ, σ∈PT and bT = X sgn(σ)σ. σ∈QT The product cT = a T bT is the Young symmetrizer of the tableau. It is a scalar multiple of an idempotent of F Sd , so that for some integer nλ depending only on the partition λ, cT cT = dλ cT . The idempotent dλ cT is, in fact, a primitive idempotent, a fact that can be demonstrated by showing that dimF cT F Sd cT = 1. Now, as desired, multiplying Uλ by one of the symmetrizers cT for λ will choose one F Sd -irreducible factor from Uλ . However, in fact, multiplying by cT will also isolate an irreducible GL(V ) module. We now define the Schur functor ST (V ) = V ⊗d ⊗F Sd F Sd cT . When F = C and V is the natural representation of GLn (C), this will in fact isolate an irreducible representation. All the ST for T a standard tableau for λ will give isomorphic representations, so in fact the notation Sλ (V ) is usually used. A good exposition of this material may be found in [2]. The representations of GL(C) obtained in this way are almost all the irreducible representations, the remaining representation being obtained by tensoring with powers of the determinant. If d < n, then they are all nonzero. They restrict to representations of SLn (C), and they correspond to irreducible representations of the simple Lie algebra sln (C). In general, the representations of a simply connected Lie group are obtained by integration from an integrable representation of the Lie algebra at the identity, and then for a non-simply connected Lie group, the representations are those of the simply connected group which are invariant under the action of the group by which one takes the quotient. We now return to the case of finite algebraic groups, when F has characteristic 0. One of the principal methods for forming irreducible characters is called Harish-Chandra induction, and this was generalized to Deligne-Lusztig induction. In Harish-Chandra induction, the representations of a Levi subgroup invariant under the action of the Frobenius map (raising every element of the matrix to the power q) are 13 extended to the corresponding parabolic subgroup, and from there induced up to the group. The induced representations are parametrized by Levi subgroups L and by cuspidal representations of L, that is, irreducible representations of L which are not induced from any Levi subgroup of L. The unipotent characters of GLn (Fq )are those associated to partitions of n in a manner similar to that we have seen in the non-finite case. Let q be a prime power, let n ≥ 0 be an and consider the group Gn = GL(n, Fq ), where q is a power of a prime p. We assume that F has characteristic `, where ` 6= q(q − 1). As with the modular theory of representations of the symmetric groups, the group algebra F Gn has a decomposition into blocks. To each block is associated a number of irreducible characters from the characteristic 0 case. A block which contains a unipotent character is called a unipotent block. There is a one-to-one correspondence between unipotent blocks of F GLn (Fq ) and blocks of Hn (F, q) [1]. The blocks correspond to the ee−1 , weights in a highest weight space for Λ0 for the affine Lie algebra A where e is the order of q in F . Chuang and Rouquier proved that the symmetrically-placed blocks along an i-string have equivalent derived categories [3]. References [1] [FB] P.Fong and B. Srinivasan, The blocks of finite general linear and unitary groups, Inventiones mathematicae, 69 (1982), 109-153. [2] [FH] W. Fulton and J. Harris, Representation Theory, Springer-Verlag, (1991). [3] [CR] J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and sl2 -categorification, Annals of Mathermatics, 167 (2008), 245298.