Representations of the Rook-Brauer Algebra

Transcription

1 Representations of the Rook-Brauer Algebra Elise delmas Department of Mathematics Macalester College St. Paul, MN Tom Halverson Department of Mathematics Macalester College St. Paul, MN June 20, 2012; revised July 20, 2012 Abstract We study the representation theory of the rook-brauer algebra RB k (x), which has a of Brauer diagrams that allow for the possibility of missing edges. The Brauer, Temperley-Lieb, Motzkin, rook monoid, and symmetric group algebras are subalgebras of RB k (x). We prove that RB k (n+1) is the centralizer of the orthogonal group O n (C) on tensor space and that RB k (n+1) and O n (C) are in Schur-Weyl duality. When x C is chosen so that RB k (x) is semisimple, we use its Bratteli diagram to explicitly construct a complete set of irreducible representations for RB k (x). Brauer algebra, rook monoid, orthogonal group, Schur-Weyl duality, seminormal rep- Keywords: resentation. 1 Introduction For k 0 and x C, the rook-brauer algebra RB k (x) is an associative algebra with 1 over C with a basis given by rook-brauer diagrams these are Brauer diagrams in which we may remove edges and a multiplication given by diagram concatenation. The rook-brauer algebra contains the symmetric group algebra, the rook monoid algebra [?], the Brauer algebra [?], the Temperley-Lieb algebra [?], the Motzkin algebra [?], and the planar rook monoid algebra [?] as subalgebras. In this paper, we define an action of RB k (n + 1) on tensor space V k where V has dimension n + 1 with basis v 0, v 1,..., v n, and we show that this action is faithful when n k. We then consider V = V(0) V(1) where V(0) = C-span{v 0 } is the trivial module for the orthogonal group O n (C), and V(1) = C-span{v 1,..., v n } is the defining n-dimensional module for O n (C). Then V k is a k-fold tensor product module for O n (C). We prove that the actions of RB k (n + 1) and O n (C) commute on V k and that, RB k (n + 1) = End On(C)(V k ), when n k. Thus, RB k (n + 1) and O n (C) centralize one another and are in Schur-Weyl duality on V k. Using this duality, we are able to show that for generic values of x C, RB k (x) is semisimple. Double centralizer theory and the combinatorics of the irreducible O n (C) modules which appear in V k tell us that the irreducible RB k (n + 1) modules are indexed by integer partitions λ r for 0 r k. Furthemore, dimension of these irreducible modules are given by paths in the Bratteli diagram, which are closely related to the vacillating tableaux studied in [?] and [?]. Supported in part by NSF Grant DMS

2 In Section 4 of this paper we explicitly construct the irreducible RB k (x)-modules for x C such that RB k (x) is semisimple. These modules have a basis indexed by paths in the Bratteli diagram, and we give an action of the generators on this basis. Our action is a generalization of Young s seminormal action of the symmetric group on standard Young tableaux. We use the analogous seminormal representations of the Brauer algebra [?], [?] and of the rook monoid algebra [?] in our construction. The rook-brauer algebra was first introduced in [?], and it is also studied in [?]. The monoid of rook-brauer diagrams (called the partial Brauer monoid) is examined in [?], where it is given a presentation on generators and relations. Representations of the rook-brauer algebra are studied independently (and simultaneously with this work) by Martin and Mazorchuk [?] using different methods. Martin and Mazorchuk also derive the Schur-Weyl duality between O n (C) and RB k (n). They define an action of the generators on tensor space and verify that the defining relations are satisfied. Our approach is different in that we give the action of an arbitrary rook-brauer diagram on tensor space and use the diagram calculus to prove that it is a representation (see Proposition??). Furthermore, Martin and Mazorchuk [?] construct Specht modules for the (integral) rook Brauer algebra, which correspond to a complete set of irreducible RB k (x) modules over C in the semisimple case. These are different from, but isomorphic to, the seminormal representations given in Section 4 of this paper. The Schur-Weyl duality results in this paper form the bulk of the work in the first author s 2012 honors thesis [?]. The ideas behind this paper began long ago in conversations between the second author, Georgia Benkart, and Cheryl Grood. It was to become a paper, BMWISH Algebras (Birman-Murakami-Wenzl-Iwahori-Solomon-Hecke Algebras), which was never written. We thank the anonymous referee for several helpful suggestions. 2 The Rook Brauer Algebra RB k (x) 2.1 Rook-Brauer diagrams A rook-brauer k-diagram is a graph consisting of two rows each with k vertices such that each vertex is connected to at most one other vertex by an edge. We let RB k denote the set of all rook-brauer k-diagrams, so that for example, RB 10 and { RB 2 =,,,,,,,,, The rook-brauer k-diagrams correspond to the set of partial matchings of {1, 2,..., 2k}. Rook- Brauer diagrams consist of vertical edges which connect a vertex in the top row to a vertex in the bottom row, horizontal edges which connect vertices in the same row, and isolated vertices which are not incident to an edge. We can count the number of rook-brauer diagrams of size k according to the number l of edges in the diagram. First choose, in ( 2k 2l) ways, the 2l vertices to be paired by an edge, and then connect these vertices in (2l)!! = (2l 1)(2l 3) 3 1 ways. Thus RB k = k l=0 }. ( ) 2k (2l)!!, (2.1) 2l so that the first few values are 1, 2, 10, 76, 764, 9496, , , ,.... 2

3 2.2 The rook-brauer algebras RB k (x, y) and RB k (x) We follow [?] and first define a two parameter rook-brauer algebra RB k (x, y). We then specialize to the one-parameter algebra RB k (x) = RB k (x, 1). Assume K is a commutative ring with 1 and that x and y are elements of K. Set RB 0 (x, y) = K1, and for k 1, let RB k (x, y) be the free K-module with basis the rook-brauer k-diagrams RB k. We multiply two rook-brauer k-diagrams d 1 and d 2 as follows. Place d 1 above d 2 and identify the bottom-row vertices in d 1 with the corresponding top-row vertices in d 2. The product is d 1 d 2 = x κ(d 1,d 2 ) y ε(d 1,d 2 ) d 3 (2.2) where d 3 is the diagram consisting of the horizontal edges from the top row of d 1, the horizontal edges from the bottom row of d 2, and the vertical edges that propagate from the bottom of d 2 to the top of d 1, κ(d 1, d 2 ) is the number of loops that arise in the middle row, and ε(d 1, d 2 ) is the number of isolated vertices that are left in the middle row. For example, if d 1 = and d 2 = then d 1 d 2 = = xy. Diagram multiplication makes RB k (x, y) into an associative algebra with identity element 1 k =. Under this multiplication, two vertical edges can become a single horizontal edge as in the example above, and a vertical edge can contract to a vertex. The rank of a diagram rank(d) is the number of vertical edges in the diagram. Therefore, rank(d 1 d 2 ) min(rank(d 1 ), rank(d 2 )). For 0 r k, we let J r (x, y) RB k (x) be the K-span of the rook-brauer k-diagrams of rank less than or equal to r. Then J r (x, y) is a two-sided ideal in RB k (x) and we have the tower of ideals, J 0 (x, y) J 1 (x, y) J 2 (x, y) J k (x, y) = RB k (x, y). Under multiplication in the partition algebra P k (x) defined in [?] and [?] isolated vertices and inner loops are treated the same. Thus, it is possible to view the rook-brauer algebra RB k (x, y) as a subalgebra of the partition algebra P k (x) by setting the two parameters equal to one-another; specifically, RB k (x, x) P k (x). For the Schur-Weyl duality between the rook Brauer algebra and the orthogonal group in Section 3, we want to specialize the second parameter to y = 1. For this reason, we consider the one-parameter rook-brauer algebra 2.3 Subalgebras RB k (x) = RB k (x, 1). (2.3) A number of important diagram algebras live in RB k (x) as subalgebras (we use RB k (x) instead of RB k (x, y) since each subalgebra either has no isolated vertices or requires that y = 1). Rook- Brauer k-diagrams having k edges are Brauer diagrams, and their K-span is the Brauer algebra B k (x). The set of diagrams of rank k is the symmetric group S k and their C-span is the symmetric group algebra KS k. The set of diagrams with no horizontal edges is the rook monoid R k and their 3

4 span is the rook monoid algebra KR k. Finally, a rook-brauer diagram is planar if can be drawn within the rectangle formed by its edges without any edge crossings. The planar Brauer diagrams span the Temperley-Lieb algebra TL k (x), the planar rook-monoid diagrams span the planar rook monoid algebra PR k, and the planar rook-brauer diagrams span the Motzkin algebra M k (x). An example of a diagram from each of these subalgebras follows: B 10 (x) S 10 TL 10 (x) R 10 M 10 (x) PR Generators and relations For 1 i k 1 define the following special elements of RB k (x) s i = 1 2 i 1 i i+1i+2 k and t i = 1 2 i 1 i i+1i+2 k and for 1 i k define p i = 1 2 i 1 i i+1i+2 k It is easy to verify that these diagrams generate all of RB k (x). In fact, RB k (x) can be generated by p 1, t 1 and s 1,..., s n 1. Furthermore, our subalgebras are generated as follows, B k (x) = s 1,..., s k 1, t 1,..., t k 1 KR k = p 1,..., p k, s 1,..., s k 1 KS k = s 1,..., s k 1 KPR k = p 1,..., p k, l 1,..., l k 1, r 1,..., r k 1 TL k (x) = t 1,..., t k 1 M k (x) = t 1,..., t k 1, l 1,..., l k 1, r 1,..., r k 1 where, for each 1 i k 1, we let l i = s i p i and r i = p i s i. A presentation of RB k (x) is given in [?] which we describe here. The symmetric group S k RB k (x) is generated by s i for 1 i k 1 subject to the following relations: (a) s 2 i = 1, 1 i k 1, (b) s i s j = s j s i, i j > 1, (c) s i s i+1 s i = s i+1 s i s i+1, 1 i k 2. (2.4) The Brauer algebra B k (x) RB k (x) contains S k and is generated by s i and t i subject to relations (??) and the following: (a) t 2 i = (x + 1)t i, 1 i k 1, (b) t i t j = t j t i and t i s j = s j t i, i j > 1, (c) t i s i = s i t i = t i, 1 i k 1, (d) t i t i±1 t i = t i, 1 i k 1, (e) s i t i±1 t i = s i±1 t i and t i t i±1 s i = t i s i±1, 1 i k 1, (2.5) 4

5 The rook monoid R k RB k (x) is generated by s i and p i subject to relations (??) and the following relations: (a) p 2 i = p i, 1 i k, (b) p i p j = p j p i, i j, (c) s i p i = p i+1 s i, 1 i k 1, (2.6) (d) s i p j = p j s i, i j > 1, (e) p i s i p i = p i p i+1, 1 i k 1. Finally, the rook-brauer algebra RB k (x) is generated by s i, t i, and p i subject to relations (??), (??), and (??) along with the following relations: (a) t i p j = p j t i, i j > 1; (b) t i p i = t i p i+1 = t i p i p i+1, 1 i k 1, (c) p i t i = p i+1 t i = p i p i+1 t i, 1 i k 1, (d) t i p i t i = t i, 1 i k 1, (e) p i t i p i = p i p i+1, 1 i k 1. (2.7) 3 Schur-Weyl Duality 3.1 Tensor powers of orthogonal group modules Let GL n (C) denote the general linear group of n n invertible matrices with entries from the complex numbers C, and let O n (C) = { g GL n (C) gg t = I } be the subgroup of orthogonal matrices, where t denotes matrix transpose and I is the n n identity matrix. We refer to [?] or [?] for standard results on the representation theory of the orthogonal group. Let V(0) = C be the 1-dimensional trivial GL n (C) module and let V(1) = C n be the defining n-dimensional GL n (C) module on which GL n (C) acts as n n matrices. Specifically, let v 0 be a basis for V(0) and let v 1,..., v n be a basis for V(1), then for g GL n (C) we have gv 0 = v 0 and gv j = n j=1 g ijv i, where g ij is the ij-entry of g. Then V = V(0) V(1) is an (n + 1)-dimensional GL n (C) module with basis v 0, v 1,..., v n. Consider the k-fold tensor product module V k = C-span { v i1 v ik i j {0,..., n} }, (3.1) which has dimension (n + 1) k and a basis consisting of simple tensors of the form v i1 v ik. An element g GL n (C) acts on a simple tensor by the diagonal action g(v i1 v ik ) = (gv i1 ) (gv ik ), (3.2) which extends linearly to make V k a GL n (C) module. The irreducible polynomial representations of O n (C) are of the form V λ where λ = (λ 1, λ 2,..., λ l ) is a partition with λ 1 + λ 2 n. In this notation V(0) = V and V(1) = V (1) = V. The Clebsch- Gordon formulas give the following tensor product formulas, V λ V = V λ and V λ V (1) = V µ, (3.3) µ=λ± and thus, V λ V = µ V µ, (3.4) 5

6 where the sum is over all partitions µ such that either µ = λ or µ = λ ±. By convention, we define V 0 = V. Let λ r denote the fact that λ is an integer partition of r, and for k 0, define Λ k = { λ r 0 r k }. (3.5) Define the Bratteli diagram B of O n (C) acting on V k to be the infinite rooted lattice with vertices on level k labeled by the partitions in Λ k and an edge between µ Λ k 1 and λ Λ k if and only if one of the following three conditions hold: λ = µ, λ = µ +, or λ = µ. Thus the first 4 rows of B are given by sum of squares V V V V V By this construction, we see that Λ k indexes the irreducible O n (C) modules which appear in V k. Furthermore, if we let m λ k denote the multiplicity of Vλ in V k, then by induction on the tensor product rule (??), we see that m λ k = (the number of paths of length k in B from Λ 0 to λ Λ k ). (3.6) 3.2 Centralizer of O n (C) on V k Let C k = End On(C)(V k ) be the centralizer of O n (C) acting on V k, so that { C k = φ End(V k ) φ(yw) = yφ(w) for all y O n (C), w V k }. (3.7) 6

7 Then by the classical double-centralizer theory (see for example [?] or [?, Secs. 3B and 68]), we know the following. Let m k,λ = mult V k(v λ ) denote the multiplicity of V λ in V k. Let dim(v λ ) = P λ (n) be the El-Samra-King polynomials (see Section 4). We have, C k is a semisimple associative C-algebra with irreducible representations labeled by Λ k. We let { M λ k λ r, 0 r k} denote the set of irreducible C k -modules. dim(m λ k ) = mλ k. We can naturally embed the algebras C 0 C 1 C 2, and the edges from level k + 1 to level k in B represent the restriction rule for C k C k+1. Therefore, B is the Bratteli diagram for the tower of semisimple algebras C k. The tensor space V k decomposes as V k = = = k r=0 λ r m λ k Vλ k P λ (n)m λ k r=0 λ r k r=0 λ r ( ) V λ M λ k Note that it follows from these expressions that (n + 1) k = as an O n (C)-module, as a C k -module, as an (O n (C), C k )-bimodule. (3.8) k P λ (n)m λ k. (3.9) r=0 λ r By general Wedderburn theory, the dimension of C k is the sum of the squares of the dimensions of its irreducible modules, k dim(c k ) = (m λ k )2. (3.10) 3.3 Action of RB k (n + 1) on tensor space r=0 λ r Let K = C. For d in the set RB k of rook-brauer k-diagrams, we define an action of d on the basis of simple tensors in V k by d(v i1 v ik ) = (d) j1,...,jk v j1 v jk, (3.11) j 1,...,j k where (d) j 1,...,j k is computed by labeling the vertices in the bottom row of d with i 1,..., i k and the vertices in the top row of d with j 1,..., j k. Then (d) j 1,...,j k = (ε) j 1,...,j k, ε d such that the product is over the weights of all connected components ε (edges and isolated vertices) in the diagram d, where by the weight of ε we mean { (ε) j 1,...,j k δ a,0, if ε is an isolated vertex labeled by a, = δ a,b, if ε is an edge in d connecting a and b, 7

8 and δ a,b is the Kronecker delta. For example, for this labeled diagram j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 j 10 d = i 1 i 2 i 3 i 4 i 5 i 6 i 7 i 8 i 9 i 10 we have d j 1,...,j k = δ j1,j 3 δ j2,i 4 δ j4,j 8 δ j5,j 6 δ j7,i 9 δ j9,0δ j10,i 8 δ i1,i 3 δ i2,i 6 δ i5,0δ i7,i 10. The representing transformation of d is obtained by extending this action linearly to V k. Define s, t : V V V V and p : V V on a simple tensors by s (v a v b ) = v b v a, t (v a v b ) = n δ a,b v k v k, k=0 p v a = δ a,0 v 0, (3.12) Then the representing action of s i, t i, and p i under (??) is given by s i id i 1 V s id k (i+1) V, 1 i < k, t i id i 1 V t id k (i+1) V, 1 i < k, (3.13) p i id i 1 V p id k i V, 1 i k, respectively, where id V : V V is the identity map. The action of the symmetric group generators s i by tensor place permutation is the same as in classical Schur-Weyl duality. The action of t i is the same as defined by R. Brauer [?] for the Brauer algebra, and the action of the p i is the same as defined by Solomon [?] for the rook-monoid. Proposition The map π k : RB k (n + 1) End(V k ) afforded by the action (??) of the basis diagrams is an algebra representation. Proof. It is possible to verify that the action of the generators (??) satisfies the defining relations (??)-(??). This is done in [?]. To get the action on a general basis diagram (??), one would then have to show that the action of the generators extends to (??). Our approach is to use the diagram calculus to show that the action on diagrams satisfies diagram multiplication. That is, for diagrams d 1, d 2 RB k we show that (d 1 d 2 ) j 1,...,j k = (d 1 ) j1,...,jk l 1,...,l k (d 2 ) l 1,...,l k. l 1,...,l k We do so by considering the edges of d 1 d 2 case by case. Case 1: Isolated vertices. If there is an isolated vertex in column r of the top row of d 1, then both (d 1 d 2 ) j 1,...,j k and (d 1 ) j 1,...,j k l 1,...,l k (d 2 ) l 1,...,l k contain δ ir,0 for all choices of l 1,..., l k. On the other hand, an isolated vertex in d 1 d 2 occurs when a vertical edge in d 1 is connected to a series of horizontal 8

9 edges (possibly 0 of them) in the middle row of d 1 d 2 which end at an isolated vertex in the middle row of d 1 d 2. For example, v a v b v a2 v a1 v a3 Sequentially label these vertices v a, v a1, v a3,..., v at, v b as shown in the diagram above. Then (d 1 d 2 ) j 1,...,j k contains δ a,0 and (d 1 ) j 1,...,j k l 1,...,l k (d 2 ) l 1,...,l k contains δ a,a1 δ a1,a 2 δ at,bδ b,0 which equals δ a,0 when a 1 = = a t = b = 0. The proof for when the isolated edge is in the bottom row of d 1 d 2 is analogous. Case 2: Horizontal and vertical edges. A vertical edge in d 1 d 2 occurs when a vertical edge in d 2 is connected to a vertical edge in d 1 by a series of horizontal edges (possibly 0) in the middle row of d 1 d 2. A horizontal edge in top row of d 1 d 2 results from a horizontal edge in t 1 or two vertical edges in d 1 connected by a series of horizontal edges in the middle row of d 1 d 2. For example, v a v a v b v a2 v a4 v a1 v a3 v a5 v a3 v a1 v a2 v a4 v a5 v a6 v b Label the top vertex in d 1 with v a and sequentially label the connected vertices with v a1, v a2,..., v at, v b as shown in the above diagrams. Then (d 1 d 2 ) j 1,...,j k contains δ a,b and (d 1 ) j 1,...,j k l 1,...,l k (d 2 ) l 1,...,l k contains δ a,a1 δ a1,a 2 δ at,b which equals δ a,b when a 1 = a 2 = = a t = b. The proof for when the horizontal edge is in the bottom row is analogous. Case 3: Loops. A loop in the middle row of d 1 d 2 results from a series of horizontal edges in the middle of d 1 d 2. For example, v a1 v a2 v a3 v at Starting with the left most middle vertex, label the vertices v a1, v a2,..., v at. When multiplying d 1 d 2 in RB k (n) the inner loop is removed and the product is scaled by n + 1 (see (??)). On the other hand, the coefficient (d 1 ) j 1,...,j k l 1,...,l k (d 2 ) l 1,...,l k contains δ a1,a 2 δ a2,a 3 δ at,a1 which is equals 1 when a 1 = a 2 = = a t is any of the n + 1 values in {0, 1,..., n} and is 0 otherwise. Proposition The representation π k : RB k (n + 1) End(V k ) is faithful for n k. Proof. Suppose there exists some nonzero y = d RB k a d d Ker(π k ). Choose a diagram d in the linear combination for y such that (i) a d 0, (ii) among the diagrams satisfying (i), d has a maximum number of vertical edges m, and (iii) among the diagrams satisfying (i) and (ii), d has a maximum number of horizontal edges l. 9

10 Now, consider the simple tensor u such that (i) v 0 is in the positions of the isolated vertices in the bottom row of d, (ii) v 1, v 2,..., v m are in the positions of the bottom vertices of the vertical edges in d, and (iii) v m+1, v m+2,..., v m+l are in the positions of the vertices of the horizontal edges in the bottom row of d such that the subscripts of the vectors in the positions of either end of a horizontal edge are the same. Similarly, consider the simple tensor u such that (i) v 0 is in the positions of the isolated vertices of the top row of d, (ii) v 1, v 2,..., v m are in the positions of the top vertices of the vertical edges in d, and (iii) v m+1, v m+2,..., v m+s are in the positions of the vertices of the horizontal edges in the top row of d such that the subscripts of the vectors in the positions of either end of a horizontal edge are the same. For example u : v 5 v 5 v 2 v 0 v 1 v 6 v 7 v 7 v 6 v 3 v 0 v 0 v 4 u : v 5 v 5 v 1 v 6 v 7 v 7 v 6 v 2 v 3 v 4 v 8 v 8 v 0 Note that the hypothesis n k guarantees that such simple tensors u and u exist. By this construction, the simple tensor u has coefficient 1 in the expansion of d u. We claim that no other diagram with nonzero coefficient in y will act on u with a nonzero coefficient of u. In order for another diagram d in y to possibly produce a nonzero coefficient of u when acting on u, d must have the same bottom row as d. This follows from the choice of d having the maximum number of vertical edges and horizontal edges and choosing distinct v i to put in each position of the edges of d. The same conditions force the top row of d and d to be the same. Therefore, d = d and thus y u 0, which contradicts the assumption that y Ker(π k ). Thus Ker(π k ) = 0 and π k is faithful. Theorem Let ρ V k : O n (C) End(V k ) denote the representation of O n (C) on V k. Then π k (RB k (n + 1)) and ρ V k(o n (C)) commute in End(V k ). Thus, π k (RB k (n + 1)) End On(C)(V k ) and ρ V k(o n (C)) End RBk (n+1)(v k ). Proof. The elements t i, s i, p j (1 i < k, 1 j k) generate RB k (n). Let g O n (C) have matrix entries g ij so that g v j = n i=1 g ij v i and g v 0 = v 0. The fact that gg T = I n since g O n (C) tells us that n l=1 g ilg jl = δ ij. From (??) and (??) we see that s i, t i act only on tensor positions i and i + 1 and p j acts only on tensor position j (i.e., these generators act as the identity elsewhere), so it is sufficient to verify that g commutes with s and t on V V and that g commutes with p on V. These are straight-forward calculations and we provide one illustrative calculation here. Let 1 a, b n. Then gt (v a v b ) = δ ab tg (v a v b ) = n g(v l v l ) = δ ab (v 0 v 0 ) + δ ab l=0 n n l=1 i=1 j=1 n n = δ ab (v 0 v 0 ) + δ ab (v i v i ) = δ ab (v i v i ), n i=1 j=1 i=1 n g ia g jb t(v i v j ) = i=0 n g ia g ib t(v i v i ) i=1 n = δ ab t(v i v i ) = δ ab (v i v i ), l=0 n g il g jl (v i v j ) 10

11 The cases where a = 0 or b = 0 are even easier to verify and they must be considered separately since g acts differently on v 0. The previous theorem gives a faithful embedding of RB k (n + 1) in End On(C)(V k ) for n k. In the next section, we prove that the dimension of the centralizer algebra equals the dimension of RB k (n + 1) and therefore RB k (n + 1) = End On(C)(V k ), n k. (3.17) 3.4 RSK insertion, paths, and vacillating tableaux For λ Λ k we define a rook-brauer path (or simply a path) of shape λ and length k to be a path of length k in the Bratteli diagram B from Λ 0 to λ Λ k. These paths exactly correspond to sequences (λ 0, λ 1,..., λ k ) of integer partitions such that (i) λ 0 =, (ii) λ k = λ, and (iii) λ i+1 is obtained from λ i by adding a box (λ i+1 = λ i + ), removing a box (λ i+1 = λ i ), or doing nothing (λ i+1 = λ i ). As an example, the following is a path of shape λ = (2) and length k = 14, (,,,,,,,,,,,,,, These paths are very closely related to the vacillating tableaux studied in [?] and [?]. Let T k (λ) denote the set of paths of shape λ and length k. Then from (??) we have T k (λ) = m λ k = dim(mλ k ) = mult V k(v(λ)). The rook-brauer diagrams in this paper are special cases of the set partition diagrams in the partition algebra defined in [?], [?]. In [?], a Robinson-Schensted-Knuth insertion algorithm is given that turns set partition diagrams to pairs of paths in the Bratteli diagram of the partition algebra. When the algorithm is restricted to rook-brauer diagrams, at each step we either add a box, remove a box, or stay the same, and thus we obtain a bijection between rook-brauer diagrams and pairs of rook-brauer paths (S, T ) of shape λ Λ k and length k. We illustrate this bijection in Example?? below. Our bijection gives us the following dimension formula, dim(rb k (x)) = RB k = k T k (λ) 2 = r=0 λ r which completes the proof of (??). ). k (m λ k )2 = dim(end On(C)(V k )), (3.18) r=0 λ r Example In this example, we illustrate the bijection of [?] that turns rook-brauer diagrams into pairs of rook-brauer paths. First we label our diagram 1,..., k in the top row and 2k,..., k +1 in the bottom row and unfold our diagram to put it in a single row: We then label each edge in the diagram by 2k i + 1, if i is the right endpoint of the edge. 11

12 If vertex i is the left endpoint of edge a, we let E i = a L, if vertex i is the right endpoint of edge a, we let E i = a R, and if vertex i is not incident to an edge, we let E i =. In this way we get an insertion sequence of the form, (E 1, E 2,..., E 2k ) = (3 L, 9 L, 1 L, 9 R, 4 L,, 5 L, 5 R, 4 R, 3 R,, 1 R ). Now let T (0) = and recursively define T (i) = E i E i RSK T (i 1), jdt T (i 1), if E i = a L if E i = a R where RSK denotes Robinson-Schensted-Knuth column insertion and jdt denotes jeu de taquin (for details on these well-known combinatorial algorithms see [?] and the references therein). In our running example, we insert and delete as follows: 1) 3 L RSK 3 7) 5 L RSK ) 9 L RSK 3 9 8) 5 R jdt 3) 1 L RSK ) 4 R jdt ) 9 R jdt ) 3 R jdt 1 5) 4 L RSK ) 1 6) ) 1 R jdt We then take the first k shapes and the last k shapes (reversed) to be our pair of paths P = (,,,,,, ), Q = (,,,,,, ). 3.5 Semisimplicity of RB k (x) for generic x C The coefficients of the relations in the presentation of RB k (x) are polynomials in x, and RB k (x) is semisimple for each x Z with x k (i.e., infinitely many). Thus, it follows from the Tits deformation theorem (see for example [?, Theorem 5.13] or [?, 68.7]) that (a) RB k (x) is semisimple for all but a finite number of x C. (b) If x C is such that RB k (x) is semisimple, then (a) Λ k is an index set for the irreducible RB k (x)-modules. (b) If M λ k is the irreducible RB k(x)-module labeled by λ Λ k, then dim(v λ ) = m λ k. 12

13 4 Seminormal Representations Throughout this section let K = C and let x C be chosen such that RB k (x) is semisimple. We explicitly construct RB k (x)-modules M λ k, for each λ Λ k. These representations have bases labeled by the paths to λ in the Bratteli diagram and they are generalizations of the seminormal representations of the symmetric group (due to Young [?]), of the Brauer algebra (due to Nazarov [?] and Leduc and Ram [?]), and of the rook monoid algebra (due to Halverson [?]). The matrices of the representations that factor through the symmetric group are orthogonal with respect to this basis. We give explicit actions of our generators p i, t i, s i on the path basis, and we show that if x is chosen so that RB k (x) is semisimple, then the modules M λ k form a complete set of pairwise nonisomorphic representations. Furthermore, as shown in (??) the seminormal bases constructed here as paths in the Bratteli diagram explicitly realize the restriction rules given by the Bratteli diagram. This is a defining feature of a seminormal basis as discussed in [?]. 4.1 El-Samra King Polynomials For a partition λ, the dimension of the irreducible O n (C) module V λ is given by the El-Samra King polynomial [?] (see also [?, (6.21)]). These polynomials occur in the matrix entries in the seminormal representations of RB n (x) that we determine below. For a partition λ let λ i denote the length of the ith column and let λ j denote the length of the jth column. Define the hook of the box at position (i, j) λ to be and, for each box (i, j) λ define d(i, j) = The El-Samra King polynomials are given by h(i, j) = λ i i + λ j j + 1. { λ i + λ j i j + 1, if i j, λ i λ j + i + j 1, if i > j. (4.1) P λ (x) = (i,j) λ and the irreducible O n (C)-module V λ has dimension given by [?] 4.2 Action of generators on paths dim(v λ ) = P λ (n). x 1 + d(i, j), (4.2) h(i, j) In this section we describe the action of each of our generators t i, s i, p i of RB k (x) on a basis indexed by the paths T k (λ) defined in??. For each λ Λ k, let { v S S T k (λ)} be a set of linearly independent vectors indexed by T k (λ), and let M λ k = C-span{ v T T T k (λ) } be a vector space spanned by these vectors. If S = (λ 0, λ 1,..., λ k ) T k (λ), then for each 1 j k define { v S, if λ j 1 = λ j, p j v S = 0, otherwise. (4.3) 13

14 Observe that the action of p j depends only on the shapes in positions i 1 and i of S. For 1 i < k, the action of the generators t i and s i depend only on the partitions in positions i 1, i, i + 1 of a path S T k (λ). For this reason, we use the notation, if S = (λ 0,..., λ k ) then S i = (λ i 1, λ i, λ i+1 ), 0 < i < k. Furthermore, we say that a pair of paths S = (λ 0,..., λ k ) and T = (τ 0,..., τ k ) are i-compatible if σ j = τ j whenever j i. That is, S and T can only differ in the ith position. We let Γ i (S) denote the set of all paths that are i-compatible with S. Note that S Γ i (S), and if T Γ i (S), then S and T have the form S = ( λ (0)..., λ (i 2), γ, α, ρ, λ (i+2),..., λ (k) ), S i = ( γ, α, ρ ), T = ( λ (0)..., λ (i 2), γ, β, ρ, λ (i+2),..., λ (k) ), T i = ( γ, β, ρ ). We will see that when s i or t i acts on a path indexed by S it stays within the span of the paths indexed by Γ i (S). For 1 i < k and S T k (λ) with S i = (γ, α, ρ) define where the constant (t) γαρ γβρ is given by t i v S = T Γ i (S) T i =(γ,β,ρ) (t) γαρ γβρ v T, (4.4) (t) γαρ γβρ = δ Pα (x 1)P β (x 1) ργ P γ (x 1) (4.5) such that δ ργ is the Kronecker delta and P λ (x) is the El-Samra-King polynomial (??). The interesting form of this matrix entry, involving dimensions of orthogonal group modules, is explained in [?]. The operator t acts as a projection onto the invariants in V V (see (??)). Leduc and Ram use the Schur-Weyl duality and a Markov trace on the centralizer algebra to prove in [?, Theorem 3.12] that in a seminormal representation any projection onto the invariants in a tensor product centralizer algebra will have this form. The fact that x 1 is used in place of x is because RB k (n) is in Schur-Weyl duality wih O(n 1). Similarly, for S T k (λ) with S i = (γ, α, ρ) define s i v S = T Γ i (S) T i =(γ,β,ρ) (s) γαρ γβρ v T. (4.6) The constant (s) γαρ γβρ is more complicated. First, we have the cases where α = γ, α = ρ, β = γ, or β = ρ. These special cases are given by the following Kronecker deltas, (s) γγρ γβρ = δ βρ, (s) γρρ γβρ = δ γβ, (s) γαρ γγρ = δ αρ, (s) γαρ γρρ = δ γα. (4.7) In particular, equation (??) says that if S and T are i-compatible and S i = (γ, γ, ρ) and T i = (γ, ρ, ρ), then s i v S = v T and s i v T = v S. and in the special case where S i = (γ, γ, γ) we have s i v S = v S. 14

15 Now, assume that α γ, α ρ, and α β. Then 1, if ρ γ, (s) γαρ ( γαρ = γαρ) ( ) 1 1 ( Pα(x 1) P γαρ) γ(x 1), if ρ = γ, ( ( γαρ) 1)( ( γαρ γαρ)+1), if ρ γ, (s) γβρ γαρ = ( γαρ γαρ) 1 Pα(x 1)Pβ (x 1) P γ(x 1), if ρ = γ, ( γαρ γβρ) with (and the following definition holds for the special case α = β), ( ) γαρ {± (ct(b) ct(a)), ρ = β ± b and α = γ ± a, γβρ = ± (x 2 + ct(b) + ct(a)), ρ = β ± b and α = γ a, (4.8) (4.9) (4.10) where λ = µ ± b indicates that the partition λ is obtained by adding (subtracting) a box a from the partition µ and where ct(a) is the content of the box a, namely ct(a) = j i, if a is in row i and column j. Remark If a segment of a path consists only of adding boxes, then the action of s i on that part of the path corresponds to Young s seminormal representation of the symmetric group S k found in [?, REF]. If the segment consists of adding boxes and doing nothing then the action of s i and p i on that part of the path corresponds to the seminormal representation of the rook monoid R k found in [?, Thm. 3.2]. If the segment consists of adding and deleting boxes (but not doing nothing), then the action of s i and t i on that part of the path corresponds to the seminormal representation of the Brauer algebra B k (x) given in Nazarov [?, Sec. 3] and Leduc-Ram [?, Thm. 6.22]. 4.3 Seminormal representations For each λ Λ k, extend the action in (??), (??), (??) linearly to M λ k and define an action of RB k(x) on M λ k by extending the action of the generators to all of RB k(x). Theorem For each k 0 and each λ Λ k, the actions of p i, t i, s i given in (??), (??), (??) extend linearly to make M λ k an RB k(x) module. Proof. To show that M λ k is an RB k(x)-module, we verify relations (??),(??), (??), (??). Many of these verifications come for free from CS n, B n (x + 1), CR n and Remark??. The generators p i, t i, s i act locally in the sense that they only care about the parts in positions i 1, i, i + 1 of the path, so (??)(b), (??)(b), (??)(b),(d), and (??)(a) all follow from the fact that the generators commute when they are working on non-overlapping parts of paths. Furthermore, we only need to check the relations when we do nothing on one of these steps (that is, λ i 1 = λ i or λ i = λ i+1 or both), since the other cases are handled in CS n, B n (x), and CR n. The symmetric group relations (??) are proven on Brauer type paths in [?] and [?]. These relations are easy to check on paths that do nothing at least once. As an example, we verify the braid relation (??)(c) on one such path. We let (δ, δ, γ, ρ) denote the path S = (λ 0,..., λ k ) such that (λ i 1, λ i, λ i+1, λ i+2 ) = (δ, δ, γ, ρ). Then, s i s i+1 s i (δ, δ, γ, ρ) = s i s i+1 (δ, γ, γ, ρ) = s i (δ, γ, ρ, ρ) = α (s) δγρ δαρ (δ, α, ρ, ρ), s i+1 s i s i+1 (δ, δ, γ, ρ) = α (s) δγρ δαρ s i+1s i (δ, δ, α, ρ) = α (s) δγρ δαρ s i+1(δ, α, α, ρ) = α (s) δγρ δαρ (δ, α, ρ, ρ). 15

16 In [?, 3.12] a general formula for action of t i is given, since t is projection onto the invariants in V V. The Leduc-Ram formula specializes in our case to?? and the relations involving only t i s follows from the general principles in [?]. Most of these relations are also easily verified directly. One subtlety is that when t i acts on a path (even on a path that has no do-nothings) the output is a sum of paths that includes paths that do nothing, and to verify the relations these do-nothing-paths need to be considered separately. As an example, we verify one of the more complicated relations (??)(e). s i t i+1 t i (γ, δ, τ, ρ) = δ γ,τ t γδγ γαγs i t i+1 (γ, α, γ, ρ) = δ γ,τ t γδγ γργs i t i+1 (γ, ρ, γ, ρ) α = δ γ,τ s i+1 t i (γ, δ, τ, ρ) = δ γ,τ α β t γδγ γργt ργρ ραρs i (γ, ρ, α, ρ) = δ γ,τ t γδγ γβγ s i+1(γ, β, γ, ρ) = δ γ,τ β α β β γ,β α α α β,α ρ t γδγ γργt ργρ ραρs γρα γβα t γδγ γβγ sβγρ βαρ (γ, β, α, ρ) (γ, β, α, ρ) Note in the first case that s γρα γβα = 0 when α = ρ, and in the second case sβγρ βαρ = 0 if β = γ. So the sum is over all paths that always add or subtract boxes in these components, and thus the two calculations are equal because they are equal in the Brauer algebra [?], [?]. It is also possible to prove this directly using a case-by-case examination of the coefficients t γδγ γργt ργρ ραρs γρα γβα and tγδγ γβγ sβγρ βαρ. The relations in (??) are straightforward. The most interesting one of them is (??)(c): s i p i (γ, α, ρ) = δ γ,α s i (γ, γ, ρ) = δ γ,α (γ, ρ, ρ) p i s i (γ, α, ρ) = β s γαρ γβρ p i(γ, β, ρ) = s γαρ γγρ(γ, γ, ρ) which are equal because s γαρ γγρ = δ α,ρ (see (??)). The relations in (??) are similarly straightforward, and we verify (??)(c) as an example: t i p i t i (γ, α, ρ) = δ γ,ρ t γαγ γβγ t ip i (γ, β, γ) = δ γ,ρ t γαγ γγγ t i (γ, γ, γ) = δ γ,ρ t γαγ γγγ t γγγ γβγ (γ, β, γ). β β Now t γαγ γγγ t γγγ γβγ = Pα(x 1)Pγ(x 1) Pγ(x 1)Pβ (x 1) P γ(x 1) P γ(x 1) = P γ(x 1) sum equals t i (γ, α, ρ). This proves that M λ k is an RB k(x) module. Pα(x 1)P β (x 1) = t γαγ γβγ and so the displayed For λ Λ k let Λ λ k 1 = { µ Λ k 1 µ = λ or µ = λ ± } so that Λ λ k 1 is the set of partitions µ Λ k 1 that are connected to λ Λ k by an edge in B. Consider RB k 1 (x) RB k (x) to be the subalgebra spanned by p i for 1 i k 1 and t i, s i for 1 i k 2. Equivalently, RB k 1 (x) is spanned by the diagrams in RB k having a vertical edge connecting kth vertex in the top row to the kth vertex in the bottom row. Now order the basis of M λ k in such a way that the paths that pass through µ Λ λ k 1 are grouped together in the ordering, and let ρλ k denote the representation corresponding to the module M λ k with respect to this basis. Then since the action of RB k 1(x) does not see the last edge in the path to λ, we have the following key property of a seminormal basis, ρ λ k (d) = ρ µ k 1 (d), for all d RB k 1(x). (4.13) µ Λ k 1 µ=λ or µ=λ± The restriction rules for RB k 1 (x) RB k (x) follow: Res RB k(x) RB k 1 (x) (Mλ k ) = 16 µ Λ k 1 µ=λ or µ=λ± M µ k 1. (4.14)

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