Journal of Algebra 238, 502᎐533Ž. 2001 doi:10.1006rjabr.2000.8613, available online at http:rrwww.idealibrary.com on Characters of the Partition Algebras Tom Halverson1 Department of Mathematics and Computer Science, Macalester College, Saint Paul, CORE Minnesota 55105 Metadata, citation and similar papers at core.ac.uk E-mail: [email protected] Provided by Elsevier - Publisher Connector Communicated by Peter Littelmann Received January 10, 2000 Frobeniuswx Fr determined the irreducible characters of the symmetric group by showing that they form the change of basis matrix between power symmetric functions and Schur functions. Schurwx Sc1, Sc2 later showed that this is a consequence of the fact that the symmetric group and the general linear group generate full centralizers of each other on tensor space. In his book, ‘‘The Classical Groups,’’ Weylwx Wy uses this duality to study representations and invariants for the general linear, orthogonal, symplectic, and symmetric groups. In 1937, Brauerwx Br analyzed Schur᎐Weyl duality for the symplectic and orthogonal groups and gave a combinatorial description of their centralizer algebras on tensor space. These centralizers are now called Brauer algebras. Only recently has Schur᎐Weyl duality been studied for the last of Weyl’s classical groups. If V s ރ r is the permutation representation for the symmetric group Sr , then its centralizer algebra has a combinatorial basis given by the collection of set partitions ofÄ4 1, 2, . , 2n . The algebra is denoted PrnrŽ.and is called the partition algebra. Since S lives inside the general linear and orthogonal groupŽ as the subgroup of permutation matrices. , the partition algebra contains the Brauer algebra and the mn symmetric group Sn Žthis is a different symmetric group which acts on V by tensor place permutations. The partition algebra appears independently in the work of Martin wxMa1᎐Ma4 and Jones wx Jo . They both study the partition algebra as a 1 Research supported in part by National Science Foundation Grant DMS-9800851 and in part by the Institute for Advanced Study under National Science Foundation Grant DMS 97-29992. 502 0021-8693r01 $35.00 Copyright ᮊ 2001 by Academic Press All rights of reproduction in any form reserved. CHARACTERS OF THE PARTITION ALGEBRAS 503 generalization of the Temperley᎐Lieb algebra and the Potts model in statistical mechanics. The algebra appears implicitly inw Ma1; Ma2, Chap. 3xwxwx and explicitly in Ma3 . Jones Jo explicitly lays out the Schur᎐Weyl wx duality between PrnrŽ.and the symmetric group S . Martin Ma3, Ma4 and Martin and Woodcockwx MW1, MW2 extensively study the structure of the ␰ ␰ g ރ general partition algebra PnŽ., with parameter . They show that ␰ ␰ wxy PnŽ.is semisimple whenever is not an integer in 0, 2n 1 , and they analyze the irreducible representations in both the semisimple and non- semisimple cases. Ramwx Ra2 uses the duality between the orthogonal group and the Brauer algebra to derive a Frobenius formula for the Brauer algebra, and uses symmetric functions to determine a recursive Murnaghan᎐Nakayama rule for Brauer algebra characters. In this paper, we determine analogues of the Frobenius formula and the Murnaghan᎐Nakayama rule for the characters of the partition algebras. The Frobenius formulas for the symmetric group and the Brauer algebra are identities in the ring of symmetric functions. Our analog lives in the character ring of the symmet- ric group. Our Murnaghan᎐Nakayama rule contains, as a special case, the Murnaghan᎐Nakayama rule for the symmetric group. The Schur᎐Weyl duality between the symmetric group and the general linear group and between the orthogonal group and the Brauer algebra each have quantum generalizationsŽ for generic values of q, seew CP, Theorem 10.2.5x and the references there. The Iwahori᎐Hecke algebra of type A is a q-generalization of the symmetric group, and it is in Schur᎐Weyl duality with the quantum general linear group on tensor space. The Birman᎐Murakami᎐Wenzl algebra is a q-generalization of the Brauer algebra, and for q generic it is in Schur᎐Weyl duality with the quantum orthogonal group on tensor space. Moreover, Frobenius formulas and Murnaghan᎐Nakayama rules for the Iwahori᎐Hecke algebrawx Ra1 and the Birman᎐Murakami᎐Wenzl algebrawx HR have been computed. However, to our knowledge, there is no q-generalization of the partition algebra. Summary of Results Ž.1 In Section 1 we develop the representation theory of the parti- tion algebra PrnŽ.using double centralizer theory and the representation theory of the symmetric group Sr . Ž.2 In Section 2, we generalize the cycle type of a permutation in the symmetric group to work for elements in the partition algebra. The ‘‘type’’ ␮ of a basis element in PrnŽ.is given by an integer partition with 0 F <<␮ F n. We show that the value of any character on a basis element d is determined by its value on d␮, where ␮ is the type of d. These special 504 TOM HALVERSON elements d␮ are analogous to conjugacy class representatives in the symmetric group. Ž.3 In Section 3, we derive a formula, ny< ␮ < ␭ ␭ r f␮ ␳ s ␹ ␥␹␳ d␮ , Ž.Ý Srn Ž.P Žr . Ž. ␭& U r <<␭ Fn relating Srncharacters and PrŽ.characters. The function f␮ is a class function on the symmetric group Sr which is explicitly computed in ␭ Theorem 3.2.2. The partition algebra ␹ Ž.d␮ can then be computed PnŽr. ny< ␮ < ␭ using the usual inner product ²rf␮, ␹ : on symmetric group charac- Sr ters. This formula is analogous to Frobenius’ formula for Sn characters. Ž.4 In Section 4, we compute an inner productŽ Proposition 4.2.1. in the character ring of Sr which allows us to derive a recursive formula ␭ Ž.Theorem 4.2.2 for ␹ Ž.d␮ . An interesting thing about this formula is PnŽr. that, like the Murnaghan᎐Nakayama rule for symmetric group characters, the recursion works by removing border strips from the partition ␭, only in this case we remo¨e border strips and then put them back on. Ž.5 For an indeterminate x, the generic partition algebra Pxn Ž .over ރ ␰ g ރ ␰ Ž.x is semisimple, and for the specialized algebra PnŽ.is semisimple whenever ␰ is not an integer in the rangewx 0, 2n y 1 . We show ␰ g ރ ␰ that for all but a finite number of , the characters of PnŽ.are the s ␰ evaluation of the corresponding PxnŽ.characters at x . We use this ᎐ fact to lift the Murnaghan Nakayama rule from PrnnŽ.to Px Ž .in Theo- rem 4.2.4. Ž.6 Let ⌶ denote the character table of the partition algebra PnŽ x. PxnŽ., that is, the table of characters which has rows indexed by the irreducible representations and columns indexed by the elements de- scribed inŽ. 1 . Using our character formula and suitable orderings on the rows and columns, we see that x⌶ . ) Pny 1Ž x. ⌶ s иии иии ,0.1 PnŽ x. Ž. 0.. ⌶ . Sn where ⌶ is the character table of S . Our rule allows us to efficiently Snn compute the values for ), and, in the lower-right corner of the table, it ᎐ specializes to the Murnaghan Nakayama rule for Sn. We give character s tables for PxnŽ.when n 2, 3, 4, 5 in Section 6. : Ž.7 There is a natural embedding Pxny1Ž.Pxn Ž.which has special properties that are exploited by Martinwx Ma3, Ma4 to construct the CHARACTERS OF THE PARTITION ALGEBRAS 505 irreducible representations of PxnŽ.. In Section 5, we prove general results about the characters of algebras with these properties. In particular, Proposition 5.1.9 tells us that the character table ⌶ must have the PnŽ x. form shown inŽ. 0.1 . The Murnaghan᎐Nakayama rule is still needed, however, to directly compute the entries from ). The general construction in Section 5 holds for the rook monoidŽ see wxMu1, Mu2, So2. , the Solomon᎐Iwahori algebraswx So1, So3, Pu, DHP , and the ‘‘rook’’ version of the Brauer and Birman᎐Murakami᎐Wenzl algebras. The construction is similar to the basic construction of Jones, which is used inwx HR to compute the characters of the Temperley᎐Lieb, Brauer, Birman᎐Murakami᎐Wenzl, and Okada algebras. 1. PARTITION ALGEBRAS In this section we give the combinatorial definition of the partition algebras and describe its irreducible representations using its duality with the symmetric group on tensor space. 1.1. Integer Partitions and Symmetric Functions A composition ␭ of the positive integer n, denoted ␭ * n, is a sequence ␭ s ␭ ␭ ␭ <<␭ s ␭ q иии q␭ s of nonnegative integers Ž.12, ,..., t such that 1 t ␭ ␭ & ␭ G ␭ G иии G ␭ n. The composition is a partition, denoted n,if 12 t. The length l Ž.␭ is the number of nonzero parts of ␭. Given two partitions ␭ ␮ ␮ : ␭ ␮ F ␭ ␮ : ␭ ␭r␮ , , we say that if iifor all i.If , let denote the skew shape given by removing ␮ from ␭. The size of a skew shape is <<<<<<␭r␮ s ␭ y ␮ . The YoungŽ. or Ferrers diagram of ␭ is the left-justified ␭ array of boxes with i boxes in the ith row. For example, Ž.5,5,3,1 s Ž.Ž.5,5,3,1 r 3,2,1 s ␭ ␭ ␭ Let miiŽ.denote the number of parts of which are equal to i.We ␭ ␭ sometimes use the notation ␭ s Ž1m1Ž . , 2m 2Ž . , . .. In this notation the partition above is given byŽ 1, 3, 52 . 506 TOM HALVERSON For a positive integer k define the power symmetric function in the variables x1,..., xr to be s k q иии q k pxk Ž.1 ,..., xr x1 xr , and for a partition ␮ define p␮ s pp␮␮иии p ␮ . 12 l Ž ␮. For a partition ␭ & r, a column strict tableau T of shape ␭ is a filling of the boxes of the Young diagram of ␭ with entries fromÄ4 1, .