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 ppиии 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, . . . , r in such a way that the entries in the columns strictly increase when read from top to bottom and the entries in the rows weakly increase when read from left to right. For a column strict tableau T, let mTiŽ.denote the number of entries of T equal to i. The Schur function can be defined as
s m1ŽT . m 2ŽT . иии m rŽT . sxŽ.1 ,..., xr Ý x12x xr , T where the sum is over all column strict tableaux of shape .
1.2. Set Partitions and Partition Algebras We consider simple graphs on two rows of n-vertices, one above the other. The connected components of such a graph partition the 2n vertices into k disjoint subsets with 1 F k F 2n. We say that two diagrams are equivalent if they give rise to the same partition of the 2n vertices. For example, the following are equivalent graphs.
We will use the term partition diagram Žor sometimes n-partition diagram to indicate the number of dots. to mean the equivalence class of the given graph. The number of n-partition diagrams which partition the 2n vertices into exactly k subsets is the Stirling number SŽ.2n, k , and the total number of n-partition diagrams is the Bell numberŽ see, for example,w Mac, I.2, Ex. 11x.
2n BŽ.2n s Ý S Ž2n, k ..Ž. 1.2.1 ks1 CHARACTERS OF THE PARTITION ALGEBRAS 507
Let x be an indeterminate. We multiply two n-partition diagrams d1 and d2 as follows:
Ž.1 Place d12above d and identify the vertices in the bottom row of d1 with the corresponding vertices in the top row of d2 . This partition diagram now has a top row, a bottom row, and a middle row. Ž.2 Let ␥ denote the number of subsets in this new partition diagram which contain vertices only from the middle row.
Ž.3 Let d3 denote the n-partition diagram obtained by using only the top and bottom rowŽ. ignoring the middle row with the imposed connec- tions.Ž Note that we have removed the ␥ disjoint subsets that were only in the middle row.. s ␥ Ž.4 The product of d12and d is then given by dd123xd.
For example,
This product is associative and is independent of the graph that we choose to represent the n-partition diagram.
Let ރŽ.x be the field of rational functions with complex coefficients in ރ an indetermine x. The partition algebra PnŽ. x is defined to be the Ž.x -span of the n-partition diagrams. Partition diagram multiplication makes PxnŽ. into an associative algebra whose identity idn is given by the partition diagram having each vertex in the top row connected to the vertex below it in the bottom row. The dimension of PxnŽ.is the Bell number B Ž2n .Žsee s ރ Ž..1.2.1 , and by convention Px0Ž.Ž.x . wx The Brauer algebra BxnnŽ.Žsee Br.Ž. is embedded in Pxas the span of the partition diagrams for which each edge is adjacent to exactly two ރ wx vertices. The group algebra Ž.xSnnof the symmetric group S is embed- ded in PxnŽ.as the span of the partition diagrams for which each edge is adjacent to exactly two vertices, one in each row. 508 TOM HALVERSON
For 1 F i F n y 1 and 1 F j F n, let
Notice that we have the relations
2 s 2 s s Ž.a AixA i,bŽ.EjxE j,cŽ.AibEE iiiq1 bi .Ž. 1.2.2
For 1 F i F n y 1 and 1 F j F n, we define
11 a s A and e s E Ž.1.2.3 iixx jj
so that aijand e are idempotent. s q g The element siicorresponds to the simple transposition s Ž.i, i 1 ¬ F F y ރ wx Sni, and the elements of the set Ä4s 1 i n 1 generate Ž.xS n. The ¬ F F y elements Ä4sii, A 1 i n 1 generate the Brauer algebra BxnŽ.. ¬ F F y F F The elements Ä4sii, b , E j1 i n 1, 1 j n generate all of PxnŽ.. To see this, think of a diagram d as living within the rectangle which has as its corners the vertices 1, n, n q 1, and 2n. Followingwx Jo , we say that a partition diagram d is planar if it is possible to draw a collection of non-intersecting closed curves inside the rectangle of d which partition the vertices of d into its equivalence classes. It is then possible to write s g w d p , where , Sn and p is a planar partition diagramŽ see Jo, x Lemma 2 for the obvious proof of this fact. . For example, the diagram d2 , above, decomposes as CHARACTERS OF THE PARTITION ALGEBRAS 509
Now, it is easy to see that each planar partition diagram can be written as F F y F F a product of bij,0 i n 1, and E ,0 j n. The planar diagram in our example can be written as
It is well known and easy to prove by induction that the number of 1 Ž.4n planar partition diagrams is the Catalan number2n q 1 2 n . Moreover, the product of two planar partition diagrams is planar, so the span of these diagrams is a subalgebra KŽ.2n, x . g ރ For each complex number , one defines a partition algebra PnŽ. over ރ as the linear span of n-partition diagrams where the multiplication is as given above except with x replaced by . The following theorem can be proved using the same argument that Wenzlwx Wz uses for the Brauer s algebra, only with the dimension of the irreducible Sr-module dimŽS . r!rh replacing the El Samra᎐King polynomials. Here h is the product of the hooks in Žseewx Mac. . wx G THEOREM 1.2.4Ž Martin and Saleur MS.Ž. . For each integer n 0, Pxn ¨ ރ ¨ ރ ¨ is semisimple o er Ž.x , and Pn Ž .is semisimple o er whene er is not an integer in the range wx0, 2n y 1.
1.3. Schur᎐Weyl Duality s ރ r ¨ ¨ ¨ Let V with standard basis 12, ,..., r be the permutation module for the symmetric group Sr . Thus,
¨ s ¨ g F F Ž.i Ži. , for Sr and 1 i r.Ž. 1.3.1
For each positive integer n, the tensor product space mn V is a module for S with a standard basis given by ¨ m ¨ m иии m ¨ , where 1 F i F r. ri12iin j g The action of Sr on a basis vector is given by
¨ m ¨ m иии m ¨ s ¨ m ¨ m иии m ¨ . 1.3.2 Ž.ii12 in Ži12. Ži . Ži n. Ž. 510 TOM HALVERSON
Number the vertices of an n-partition diagram 1, 2, . . . , n from left to right in the top row and n q 1, n q 2,...,2n from left to right in the bottom row. For each n-partition diagram d and each integer sequence F F i12,...,i nkwith 1 i r, define ¡ s 1, if i jki whenever vertex j is ␦ i1 ,...,i n s~ Ž.d i q ,...,i Ž.1.3.3 n 12n ¢ connected to vertex k in d, 0, otherwise.
g mn Define an action of a partition diagram d PrnŽ.on V by defining it on the standard basis by
i1 ,...,i n ¨ m ¨ m иии m ¨ d s ␦ d i ,...,i ¨ m ¨ m иии m ¨ . Ž.ii12 innÝ Ž.nq 12n iq1 inq22 in inq12,...,i n Ž.1.3.4
wx THEOREM 1.3.5Ž Jones Jo.Ž. . Srn and P r generate full centralizers of each other in EndŽmn V .. In particular, for r G 2n, Ž.a P Ž. r generates End Žmn V .Ž, and when r G 2n, Pr.( nSr n End Žmn V .. Sr Ž.b S generates End Žmn V .. rPnŽr. Proof. We have A g End Žmn V . if and only if y1A s A as opera- Sr tors on mn V for each g S . Let A s w Ai1, i2 ,...,i n x. Since V is the rinq 1, i nq22,...,i n y 1 permutation representation for S r , A has matrix w Ži1., Ži2 .,..., Ži n. x A . Hence the entries of the matrix A are equal for Ži nq 1., Ži nq22.,..., Ži n. all coordinates in the orbit of . If we collect the i12, i ,...,i 2n according to those that have an equal value, we form a partition ofÄ4 1, . . . , 2n into subsets, and the indices in each subset will always have equal value in an orbit of . Given an equivalence relation ; which partitions the setÄ4 1, 2, . . . , 2n n into at most r subsets, define T;g EndŽm V . by
1, if i s i whenever j ; k i12, i ,...,i n s jk Ž.T; i , i ,...,i nq 1 nq22n ½ 0, otherwise.
Then, the set T; ,as ; ranges over all partitions ofÄ4 1, 2, . . . , 2n into at most r subsets, forms a basis of End Žmn V .. Moreover, this is exactly the Sr matrix of the corresponding partition diagramŽŽ see 1.3.3 .. . When r G 2n, ; all equivalence relations occur, and the representation of PrnŽ.on EndŽmn V .Žis faithful. Part b. follows since these centralizer algebras are
semisimple and because PrnrŽ.is the centralizer of S . CHARACTERS OF THE PARTITION ALGEBRAS 511
1.4. Irreducible Representations of PnŽ. r
The irreducible representations of Sr are indexed by the partitions & r, and we denote them by S . The permutation module V decomposes into irreducibles as V ( SŽr. [ SŽry1, 1., where SŽr . is the trivial module for
Sr . The rules for Kronecker products of these special symmetric group modules are knownŽ for example they can be derived fromw Mac, I.7, Ex. x & m 23Ž. d. . For r, the decomposition of S V into irreducible Sr mod- ules is S m V ( [ V , Ž.1.4.1 sŽ.yq where the sum is over all & r such that is obtained by removing a box from to get a partition y and then adding the box back to y to get . The Bratteli diagram for r s 6 is shown in Fig. 1.
The partition algebra Pxny1Ž.is embedded in Pxn Ž.by adding a hori- zontal edge connecting the nth and the 2nth dots. The lines in the Bratteli
diagram indicate the branching rules under the restriction from PrnŽ.to wx Prny1Ž.. Martin Ma4 shows that the restriction rules are the same for PxnnŽ.and, in the semisimple cases, P Ž.. From double centralizer theoryŽ see, for examplewx CR, Sect. 3D. , we have the following remarkable consequences of Theorem 1.3.5:
Ž.1 The irreducible representations of Prn Ž.are indexed by the same mn set that indexes the irreducible Sr-modules in V. UsingŽ. 1.4.1 , we see that this is the set $ s & ¬ <<UUF s r PrnŽ. Ä4r n , where Ž.Ž.1 . 1.4.2
FIG. 1. The Bratteli diagram for PnŽ.6. 512 TOM HALVERSON
U Here is the partition obtained from by removing its first part 1, i.e.,
Ž.$2 Denote by M the irreducible representation of PrnŽ.indexed by g mn PrnŽ.. The dimension of M equals the multiplicity of S in V and thus is the number of paths from the top of the Bratteli partition diagram to . In row n s 3 of Fig. 1, these dimensions are 5, 10, 6, 6, 1, 2, 1 Ž.reading left to right . Moreover, 52q 10 22222q 6 q 6 q 1 q 2 q 1 s 203 which is the Bell number BŽ.6. mn = Ž.3 The decomposition of V as a bimodule for SrnPrŽ.is
mn ( m V [$ S M , Ž.1.4.3 g PrnŽ.
$ G ¬ g and, when r 2n, Ä4M PrnŽ. is a complete set of irreducible PrnŽ.-modules. Remark 1.4.4. We have two symmetric groups acting on mnV. The : ރ group Sr GLŽ.r, acts on the left by the tensor product of its permuta- : tion representation, and the group SnnPrŽ.acts on the right by tensor place permutations. Remark 1.4.5. In the classical casewx Sc1, Sc2 , V s ރ r is the standard representation of GLŽ. r, ރ where g g GL Ž. r, ރ acts by matrix multiplica- ރ tion on the left. Restriction from GLŽ. r, to Sr gives the permutation ރwx: representationŽ. 1.3.1 of Srn. The centralizer algebra is S PrnŽ.with the action given byŽ. 1.3.4 . When r G n,
mn V ( [ V m S , Ž.1.4.6 &n
where V is a GLŽ. r, ރ -module of highest weight and S is an irre-
ducible Sn-module. CHARACTERS OF THE PARTITION ALGEBRAS 513
1.5. Irreducible Representations of PnŽ. x $ G When r 2n, the index set PrnŽ.defined inŽ. 1.3.8 is in bijection with the set $ s s & ¬ <<U F PˆnnP Ž.2n Ä42n n .Ž. 1.5.1 $ g y If PrnŽ., the bijection is given by subtracting r 2n from 1. Since G < U < F y G y y s y r 2n and n, we see that 12Ž.r n n r 2n, and thus this subtraction will always give a partition of 2n. Martinwx Ma3 shows that, in the semisimple cases, the irreducible & representations of PxnnŽ.and P Ž .are indexed by the partitions k, F F 0 k n. These partitions are also in bijection with Pˆn. Indeed, they are U exactly the partitions inŽ. 1.5.1 and Ž. 1.4.2 . We get from Martin’s y partitions to those in Pˆn by adding a top row with 2n k boxes. The partitions in Pˆn are better suited for the recursive algorithm that we derive in Section 4. For g Pˆ , let denote the irreducible character of P Ž. corre- nPnŽ . n sponding to M . The proof of the next proposition is identical to that of Corollary 2.4 inwx Ra2 .
PROPOSITION 1.5.2. For all but a finite number of g ރ, the character of P Ž. equals the character of PŽ. x e¨aluated at x s . PnŽ . nPnŽ x. n
2. CONJUGACY CLASS ANALOGS
In this section, we divide the partition algebra diagrams into classes on which characters are constant. These are analogs of the conjugacy classes, determined by cycle type, in the symmetric group.
2.1. Standard Elements ␥ s ) ␥ s иии Let 1 1, for t 1 let ttssy1 ty21s , and let E 1be the 1-parti- tion diagram with no edges. Thus,
m If d1212and d are n and n -partition diagrams respectively, then d12d is q the Ž.n12n -partition diagram obtained by placing d2to the right of d1. 514 TOM HALVERSON
s * For Ž.1,..., l k, define
␥ s ␥m иии m ␥ g S , 1 l k and for a composition with 0 F << F n, define d s ␥ m EŽny< <. s ␥ m E m иии m E g Px. 2.1.2 1 ^`_11nŽ. Ž . n y << times g & The cycle type of a permutation Sn is the partition Ž. n given g by the lengths of the disjoint cycles in . Each Sn is conjugate to ␥ ␥ ¬ & Ž ., and Ä4 n is a complete set of Sn-conjugacy class representa- tives.
2.2. ‘‘Cycle Type’’ in the Partition Algebras g Let d PxnŽ.be a partition diagram. The vertices of d are arranged in n columns of size 2. Connect each vertex in d to the other vertex in its column with a dotted line. The connected components of this new diagram partition the vertices into disjoint subsets, called blocks. Vertices in the same column are always in the same block, and there are no connections between disjoint blocks. The following diagram consists of four blocks. The first is on columns 1, 2, 3, 5, the second on columns 4, 6, 8, the third on column 7, and the fourth on columns 9, 10.
A vertex d is isolated if it is incident to no edges, and an edge in d is horizontal if it connects vertices in the same row. If a vertex is adjacent to two vertices in the opposite row, then those two vertices are adjacent by a horizontal edge. Thus, if d has no isolated vertices and no horizontal g edges, then d Snn. Furthermore, a single block in an S diagram is a cycle. We assign a nonnegative integer to each block, called the block type, using the following algorithm:
Ž.1 While the block is not E1 or a permutation diagram, do the following: Ž.i If the block has an isolated vertex ¨, then remove the column containing ¨ from the blockŽ removing the edges incident to the other vertex in the same column as ¨ .. CHARACTERS OF THE PARTITION ALGEBRAS 515
Ž.ii If the block has a horizontal edge, then Ž.a connect the corresponding vertices in the opposite row by an edge Ž.b add any new edges that are now implied by transitivity Ž.c remove one of these two columns from the diagram.
Ž.2 If the remaining block is E1, then it has block type 0. Otherwise, the remaining block is a permutation diagram and so it is a cycle. The block type in this case is the length of this cycle, i.e., the number of columns remaining in the block. For example, we apply the algorithm to the first block in the example above:
and this block has type 1. The sequence of block types of d forms a partition s Ž0,1,...,m 0 m1 . where mi is the number of blocks of type i. Note that
y << y s n m0 Ž.the number of columns removed in the algorithm .
In our example, the blocks have type 1, 3, 0, and 1, respectively, and so the block type of the diagram isŽ 0, 12 , 3. . Here n s 10 and 10 y 5 y 1 s 4. Three columns were removed from the first block in the illustration above, and we remove 1 column when determining the type of the last block. X g X If d, d PxnŽ., then we say that d is conjugate to d if there exists g X s y1 Sn such that d d . The following properties are easy to verify: Ž.1Ifd is a partition diagram, then dy1 is the partition diagram given by rearranging the vertices of d, in both the top and bottom row, according to . y1 s Ž.2If is any character of Pxn Ž ., then Ž d .Ž.d . m m Ž.3 d12d is conjugate to d21d . Ž.4 y1 d has the same block type as d. 516 TOM HALVERSON
g G G иии G Ž.5Ifd Pxn Ž .has blocks of size k12k k l , then there g exists Sn so that
y1 s m m иии m d b12b bl , where each b consists of a single block of size k in PxŽ.. Simply choose iiki to rearrange the vertices so that those in the largest block come first, those in the next largest block come next, and so onŽ breaking ties arbitrarily. . ␥ Ž.6 Each cycle of length t is conjugate to t.
PROPOSITION 2.2.1. If d is a partition diagram with block type s m 0 m1 Ž0,1,....Žand is any character of Pn x., then
yŽ ny< s ␥ m mŽny< <. where d E1 is a standard diagram defined in Ž.2.1.2 . Proof. FromŽ.Ž. 1 ᎐ 5 above, we see that we can work on the individual blocks of d independently, and thus we assume that d consists of a single block. If there is only one column in d, then we are done, since the two possible partition diagrams in this case are of the formŽ. 2.1.2 . Thus, we assume that d has more than 1 column. Ž.iIfd has an isolated vertex at position i in the top row, then the vertex below it is not isolated, since d is a block with more than one column. Thus s s X Ediixd and dE d , X where d is the same diagram as d only the edges incident to the ith vertex in the bottom row have been removed. By permuting the ith column to the X Y m Y end, d is conjugate to d E1 where d is the diagram obtained from d by removing the ith column. Thus, 111Y Ž.d s ŽEd .s ŽdE .s Žd m E .. xxxii 1 The symmetric argument is used when the isolated dot is in the bottom row. CHARACTERS OF THE PARTITION ALGEBRAS 517 Ž.ii If d has a horizontal edge in the top row connecting vertices i and j, define Then X bd s d and db s d , X where d has an isolated vertex in the bottom row of the jth column. Now s 1 XXm caseŽ. i applies to get Ž.Ž.d x d E1 where d is obtained from d by connecting the ith and jth vertices in the bottom row, connecting any new edges implied by transitivity, and then removing the jth column. Again, the symmetric argument is used for horizontal edges in the bottom row. Each application ofŽ. i or Ž ii . removes one column from the diagram and adds E1 at the end. Eventually we reach a permutation diagramŽ no ␥ isolated vertices and no horizontal edges. , which is conjugate to a cycle t , g ރ / or we reach E1. The argument is the same for PnŽ., , 0, only with replacing x. 3. THE FROBENIUS FORMULA For a partition & r, let denote the irreducible character of the Sr & < U < F symmetric group Sr corresponding to S .If r and n Žsee Ž..1.4.2 , then let P Žr. denote the irreducible character of the partition n algebra PrnŽ.corresponding to M . 3.1. The Frobenius Formula for the Symmetric Group As in Subsections 1.3 and 1.4, let V s ރ r be the natural representation ރ mn mn of GLŽ. r, , and let V be its n-fold tensor product. Let Sn act on V by the action defined inŽ. 1.3.4 . Define the bitrace of g = g GLŽ. r, ރ = mn Sn on V to be < btrŽ. g, s g ¨ m иии m ¨ ¨ m иииm¨ ,Ž. 3.1.1 Ý Ž.ii1 n ii1 n i1,...,i n where gŽ.¨ m иии m ¨ < ¨ m иииm¨ denotes the coefficient of ¨ m иии m¨ ii1 n ii1 n ii1 n in the expansion of gŽ.¨ m иии m ¨ . Since the actions of GLŽ. r, ރ and ii1 n Sn commute, the bitrace is a trace in both components, and thus it is sufficient to compute the bitrace on conjugacy class representatives. 518 TOM HALVERSON g ރ ␥ g & For g GLŽ. r, with eigenvalues x1,..., xr and Sn with n as inŽ. 2.1.2 , Schurwx Sc1, Sc2 , proved that the bitrace is given by the power symmetric function, ␥ s btrŽ. g, px Ž1 ,..., xr ..Ž. 3.1.2 The proof ofŽ. 3.1.2 is a straightforward computationŽ seewx Ra2. . The irreducible character of GLŽ. r, ރ corresponding to the highest weight module V is the Schur function s, so by computing the bitrace on both sides ofŽ. 1.4.6 we obtain Frobenius’ formula px ,..., x s sx ,..., x ␥ . 3.1.3 Ž.Ž.Ž.Ž.1 r Ý 1 rSr &n 3.2. A Frobenius Formula for the Partition Algebras & * F F g g Let r and m with 0 m n. Then let Sr , and let d PrnŽ.be defined as inŽ. 2.1.2 . Since the actions of Srnand PrŽ.commute n on m V, we define the bitrace of btrŽ. , d exactly as in Ž. 3.1.1 only with replacing g and d replacing ␥. Then, computing the trace on the right-hand side ofŽ. 1.4.3 gives btr , d s Ž. d .Ž. 3.2.1 Ž.Ý$ SPrnŽ r . Ž. g PrnŽ. n We now compute btrŽ. , d directly on m V. g s THEOREM 3.2.2. Let Sr and Ž.1,..., l be a composition F << F = g = mn ¨ with 0 n. The bitrace of d SrnPronŽ. Visgi en by ny< < ny< < btr , d s r f s r f f иии f , Ž. Ž.12 Ž. Ž.l Ž. s ¨ where, if Ž.12, ,... is the cycle type of and k is a positi e integer, ¨ then fkŽ.is computed by any of the following equi alent formulas: l Ž . 2 y1 Ž.a f Ž .s Ý s p Ž1, , ,..., i ., ki1 k ii i s и ¨ ¨ Ž.b fkd Ž . Ý ¬ kdd m Ž .Žsumming o er the di isors d of k., s k s k Ž.c fkV Ž . Ž .Žthe number of fixed points of ., s 2 i r j ¨ such that jde is a primiti e jth root of unity, m Ž.is the number of parts of equal to d, and V is the character of the permutation representa- tion V of Sr . g Proof. If Sr , d11is an n -partition diagram, and d 22is an n -parti- m s = tion diagram, then btrŽ.Ž.Ž., d12d btr , d1 btr , d 2where d1 CHARACTERS OF THE PARTITION ALGEBRAS 519 mn1 m mn2 acts on the first V and d2 acts on the last V. It follows from Ž.2.1.2 that ny<< btr , d s btr , ␥btr , ␥ иии btr , ␥ btr , E . Ž. Ž.Ž.12 Ž.Ž.l 1 We compute the value of btrŽ., E1 on V directly. FromŽ. 1.3.4 , we see ¨ s r ¨ that i E1 Ý js1 j, and thus rrrrrr s ¨ <<¨¨s ¨ s ¨ < ¨s s btr Ž., E1 ÝÝÝÝÝÝi E1 iij Ž j. i1 r. is1 is1 js1 is1 js1 is1 s ␥ Define fkkŽ.btr Ž, ., and we have proved ny< < f s r f иии f . Ž.1 Ž.l Ž. It remains to compute fkŽ.. : ރ Ž.a Using Schur’s resultŽ. 3.1.2 restricted to Sr GLŽ. r, , we have ␥ ␥ s btr Ž.Ž , kkpx1 ,..., xr ., ␥ where x1,..., xr are the eigenvalues of . The eigenvalues of a j-cycle in jy1 g Srjare the jth roots of unity 1, ,..., j, and Sris a disjoint g product of cycles. Thus, the eigenvalues of Sr are the union of the jth roots of unity as j ranges over the lengths of the cycles in . PartŽ. a then s q follows from the fact that pxkŽ.Ž.1,..., x a, y1,..., ybkpx1,..., x a pykŽ.1,..., yb . s 2 i r j Ž.bIfj e , then jy1 s q k q 2 k q иии q kŽ jy1. pkjŽ.1, ,..., j 1 j j j kj y 1 s j k y j 1 s j,ifj divides k, ½ 0, otherwise. Thus, the only nonzero summands which appear inŽ. a are when i divides k, and in that case the value is i. We can then sum the divisors d of k, and they appear with multiplicity mdŽ.. k k Ž.c The character value V Ž . is the number of fixed points of , and the fixed points of k are the elements of the d-cycles of where d divides k. PartŽ. c then follows fromŽ. b . 520 TOM HALVERSON PuttingŽ. 3.2.1 and Theorem 3.2.2 together gives our Frobenius formula F << F for the characters of PrnŽ.. For a composition with 0 n, we have ny< < r fŽ. s d Ž. , for all g S .Ž. 3.2.3 Ý$ PnrŽr. Ž.Sr g PrnŽ. For each composition , the function f is a class function on Sr . Let ރ RSŽ.r denote the -vector space generated by the class functions of Sr . The irreducible characters of Srrform a basis of RSŽ., and the Frobenius formulaŽ. 3.2.3 says that the character Ž.d is the coefficient of PnrŽr. S ny< < when r f is expanded in terms of the irreducible Sr characters. The module RSŽ.r carries a scalar product² , : which is defined by 1 s ␥ ␥ g ²:g, h Ý g Ž.Ž. h , for g, h RSŽ.Ž.r , 3.2.4 &r z s l Ž . m iŽ . ␥ where z Ł is1 imiŽ.! is the size of the centralizer of in Sr . With respect to this product the irreducible characters of Sr are orthonor- mal, and so we have the following corollary of our Frobenius formula. COROLLARY 3.2.5. Let be a composition with 0 F << F n, and let U & r with < < F n. Then ny< < ny< < ds ²rf, : s rf² , :. PnrrŽr. Ž. S S 4. THE MURNAGHAN᎐NAKAYAMA RULE The Murnaghan᎐Nakayama rule is a recursive rule for computing characters of the symmetric group. It is derived by expanding polynomials in the ring of symmetric functions. In our work here, we must derive the rule by working in the character ring of the symmetric group. 4.1. The Murnaghan᎐Nakayama Rule for the Symmetric Group Two boxes in a skew shape r are adjacent if they share a common edge, and r is connected if you can travel from any box to any other via a path of adjacent boxes. A skew shape r is a border strip if it is CHARACTERS OF THE PARTITION ALGEBRAS 521 connected and it does not contain any 2 = 2 blocks of boxes. The following figures illustrate the three border strips of size 4 in s Ž.4, 4, 3, 1 . The height of a border strip is given by htŽ.r s ࠻ Žrows occupied by r.y 1. The heights of the border strips in the figures above are 1, 1, and 2, respectively. For a proof of the next theorem, seewx Mac, I.7, Ex. 5 . ᎐ & THEOREM 4.1.1Ž. The Murnaghan Nakayama Rule for Sn . Let n, * n, and let be obtained from by remo¨ing a part of size k. Then the irreducible character ␥Ž. is gi¨en by Sn htŽ ryk . yk ␥ sy1 ␥ , SnnŽ.Ý Ž .S yk Ž. yk: where the sum is o¨er all partitions yk : such that ryk is a border strip of size k. 4.2. A Murnaghan᎐Nakayama Rule for the Partition Algebras If ␦ is a character of S , and k is a positive integer, then f ␦ is the Srr kSr class function which takes the value f Ž.␦ Ž. on g S . The next kSr r proposition gives the expansion of f Ž.␦ Ž. in terms of the basis of kSr irreducible characters. It is the analogue of the symmetric function identi- tiesw Mac, I.3, Ex. 11Ž. 2xwx for the symmetric group and Ra2, Theorem 6.8 for the Brauer algebra. ␦ & PROPOSITION 4.2.1. Let , r. Using the inner product ²:, on Sr characters Ž.3.2.4 , we ha¨e ryd ␦ryd ²:f ␦, sy1 htŽ .Žy1,ht . kSrr S ÝÝŽ. Ž. d¬k yd: yd:␦ where the outer sum is o¨er all di¨isors d of k and the inner sum is o¨er all partitions yd & Ž.r y d such that yd : , yd : ␦ and ryd and ␦ryd are border strips of size d. If no such yd exists, then the inner product is zero. 522 TOM HALVERSON Proof. ByŽ. 3.2.4 and Theorem 3.2.2, we have ␦ ␦1 ²:f , s f ␥ ␥ kSrr S Ý kŽ. Sr Ž.S r Ž. &r z dmd Ž. ␦ s ␥␥ . ÝÝ SrrŽ.S Ž. d¬k &r z The nonzero terms of the inner sum occur when & r has at least one part of size d. These partitions are in bijection with partitions of the form ␣ q Ž.d , where ␣ & Žn y d .and ␣ q Ž.d is the partition of r obtained by adding a part of size d to ␣. Moreover, for partitions of this form, we have s ␣ q z␣qŽd. dmdŽ Ždz ..␣ Žsee Ž 3.2.4 .. , which implies that ␣ q dmd Ž.Ž.d 1 s . zz␣qŽd. ␣ Thus, ␦ ␦1 ²:f , s ␥␣q ␥␣q . kSrr S ÝÝ SrŽ.Ž.Žd. Sr Ž d. d¬k ␣&Ž.ryd z␣ ᎐ We now apply the Murnaghan Nakayama rule for Sr Ž.Theorem 4.1.1 to ␦ remove a part of size d from ␥Ž.␣q and ␥Ž.␣q : SrrŽd. S Ž d. ² f ␦ , : kSrr S 1 s ÝÝ d¬k ␣&Ž.ryd z␣ htŽ␦r␦yd .Žht ryd . ␦yd yd = y1 y1 ␥␥␣ ␣ ÝÝŽ. Ž.Sry dr Ž.S yd Ž. ␦yd:␦yd: ␦r␦yd ryd syÝÝ ÝŽ.1 htŽ .Ž Ž.y1 ht . d¬k ␦yd:␦yd: ␦yd yd = ␥␣ ␥␣ . Ý Sry drŽ.S yd Ž. ␣&Ž.ryd By orthogonality of Sryd characters, the innermost sum is equal to ␦yd yd ² , : s ␦␦yd ␦yd , where ␦ yd yd is a Kronecker delta. Thus, we SSry dryd , , must have yd s ␦yd to get a nonzero inner product, and the proposition is proved. CHARACTERS OF THE PARTITION ALGEBRAS 523 The following theorem is our analogue of the Murnaghan᎐Nakayama U rule. For a partition , recall the definition of inŽ. 1.4.2 . Since we will be using recursion on n, we will sometimes add the subscript n to the definition of d fromŽ. 2.1.2 s s ␥ m mny< < * d, n dE1 , where n. U THEOREM 4.2.2. Let & r with < < F n. Let be a composition with 0 F << F n, and let be the composition obtained from by remo¨ing a part of size k. Then htŽ ryd .Žht ␦ryd . ␦ d sy1 y1 d y , PnŽr. Ž., nPÝÝŽ. Ž. nyk Žr . Ž., n k d¬k ␦sŽ.yd qd U <<␦ Fnyk where the outer sum is o¨er all di¨isors d of k and the inner sum is o¨er all U ␦ & r with <␦ < F n y k such that ␦ is obtained from by remo¨ing a border strip of size d to obtain the partition yd and then adding a border strip of size dtoyd to obtain ␦ s Žyd .qd. Proof. Since * Ž.n y k , our Frobenius formulaŽ. 3.2.3 gives the following identity in RSŽ.r : Žnyk.y< < ␦␦ rf s d y . Ý Pny krŽr. Ž., n kS ␦& U r <<␦ Fnyk Using Proposition 4.2.1 and Corollary 3.2.5, we have ny< < ␥s ²rf, : PnrŽr.Ž. S Žnyk.y< < s ²:rff , kSr ␦ ␦ s d y ²: f , Ý Pny krrŽr. Ž., n kSkS ␦& U r <<␦ Fnyk ␦ htŽ ryd .Žht ␦ryd . s d y y1 y1 ÝÝPny k Žr. Ž., n k ÝŽ. Ž. ␦&rd¬k yd: <<␦ U F y n k yd:␦ htŽ ryd .Žht ␦ryd . ␦ sy1 y1 d y . ÝÝŽ. Ž. Pny k Ž r. Ž., n k d¬k ␦sŽ.yd qd U <<␦ Fnyk The last equality follows from changing the order of summation and observing that the inner product ² ␦ f , : is zero unless ␦ can be Skrr S obtained from by removing and then adding back a border strip of size d. 524 TOM HALVERSON U COROLLARY 4.2.3. If & r with < < F n and is a composition with 0 F << F n, then ny< < << ny< < ny< < ny< < ds ²rf, : s rf² , : s r ␥ , PnrrŽr. Ž. S SP< <Ž r.Ž. where the first and the third equality follow from Corollary 3.2.5. U If << - < <, then we recursively apply Theorem 4.2.2 until we have removed all of the parts of . We are left with a sum of characters of the form ny< < ␦ s ␦ mŽny< <. s ²:ny< < ␦ s r ,ifŽ.r , P y < <Žr. Ž.E1 r , S n r ½ 0, otherwise, where the first equality comes from Corollary 3.2.5 and the second equality comes from the fact that the only character that is constant on all Sr Žr. conjugacy classes is the trivial character S which is 1 on all conjugacy U r classes. The partitions ␦ have minimal <␦ <, if at each step in the recursion ␦ U ␦ we remove a strip of size i from and add it to the first row of .In UU this case, we will have removed << boxes from to obtain ␦ . However, U U < <<<) ,so <␦ < ) 0. In particular ␦ / Ž.r , and so the above inner product is 0 for each summand. ForŽ. b , note that in Theorem 4.2.2, if & n, then when we remove and add a border strip of size d ¬ k from to obtain ␦, we must have U <␦ < F n y k. We are thus forced to remove a border strip of size k from yd and place it at the end of the first row in ␦. The strip we add to ␦ has weight 1, and so we only consider the weight of removing a border strip of yd ᎐ size k from . This is precisely the Murnaghan Nakayama rule for Sn Ž.Theorem 4.1.1 . << s s ␥ ForŽ. c we see that if n, then byŽ. a we have P Žr .Ž.d P Žr .Ž. . U nn If & r with < < s n y l , then since r G 2n, we have y G y y y y s y q G 12^`_^`_r Ž.Ž.n l n l r 2n 2 l 2 l , G 12 so a border strip in that contains boxes from 12and has at least CHARACTERS OF THE PARTITION ALGEBRAS 525 2 l q 2 boxes. Suppose that d divides k and that we remove a border strip of size d U from and add it back to obtain ␦, with <␦ < F n y k. Assume that the G border strip that we remove contains boxes from both 12and .If k l , U then we must move at least k y l boxes from up to the first row of ␦. U But then d s Ž࠻ boxes removed from .Žq ࠻ boxes removed from G y q q s q q ) 1.Žk l .Ž2 l 1 .k l 1 k, a contradiction since d di- ) G y G ) vides k. On the other hand, if l k, then d 122 l k, also a contradiction. Thus, we can not remove a strip from that contains boxes U from both 12and , and in particular, we remove strips from 1and independently. An entirely similar argument shows that we can never add back a strip ␦ ␦ U G of size d that lives in both 1 and .Asr 2n varies, the set of possible U remains constant, only the length of the first row varies. Moreover, when r G 2n, there is only one way to remove or add a strip of size d from or to the first row. Now we use Corollary 4.2.3 to lift our character formula to PxnŽ.. U THEOREM 4.2.4. Let & r with < < F n. Let be a composition with 0 F << F n, and let be the composition obtained from by remo¨ing a part of size k. Then ny< < << htŽ ryd .Žht ␦ryd . ␦ ␥ sy1 y1 ␥ , PnŽ x.Ž. ÝÝŽ.Ž. Pnyk Ž x.Ž. d¬k ␦sŽ.yd qd U <<␦ Fnyk where the outer sum is o¨er all di¨isors d of k and the inner sum is o¨er all U ␦ & r with <␦ < F n y k such that ␦ is obtained from by remo¨ing a border strip of size d to obtain the partition yd and then adding a border strip of size dtoyd to obtain ␦ s Žyd .qd. << y y ny< < htŽ r d .Žht ␦r d . ␦ fx s x y1 y1 ␥ , Ž. ÝÝŽ.Ž. P< Since both fxŽ.and Ž.d are rational functions in x which agree at PnŽ x. an infinite number of points, then by Proposition 1.5.2 they must be equal everywhere. 5. BASIC CONSTRUCTION wx To construct irreducible representations of PxnŽ., Martin Ma3 exploits : : : иии the properties of Px012Ž.Px Ž.Px Ž. as a tower of semisimple algebras. In this section, we derive general results about the characters of algebras which have these basic properties. Other algebras of this form are the rook algebrawx Mu1, Mu2, So2 and its Iwahori᎐Hecke algebraw So1, So3, Pux and the ‘‘rook’’ versions of the Brauer and Birman᎐Murakami᎐ Wenzl centralizer algebras. Frobenius formulas and Murnaghan᎐Naka- yama rules for the rook monoid and its Iwhahori᎐Hecke algebra are determined inwx DHP . 5.1. Basic Construction Let A : B be an inclusion of finite-dimensional, semisimple algebras with 1 over ރ. Let 0 / e g B and assume that the following properties hold: Ž.a e2 s e, Ž.b ea s ae, for all a g A, Ž.5.1.1 Ž.c A ( Ae s eBe via the map a ¬ ae for all a g A. PropertyŽ. c allows us to define a map, : B ª A determined by ebe s Ž.befor all b g B.Ž. 5.1.2 PROPOSITION 5.1.3. Let B be a semisimple algebra with 1 and let e g B such that e / 0 and e2 s e. View Be as a module for BeB by multiplication on the left and as a module for eBe by multiplication on the right. Then ( ( BeB End eBeŽ.Be and eBe End BeBŽ.Be . Proof. Left multiplication by BeB commutesŽ. by associativity with : : right multiplication by eBe, so we have BeB End eBeŽ.Be and eBe g g s End BeBŽ.Be . Suppose that End BeBŽ.Be . Then Ž.e Be so Ž.e CHARACTERS OF THE PARTITION ALGEBRAS 527 Ž.ee. Since be s be1 g BeB, we have Žbe .s Žbee .s be Ž.e s bee Ž.ee, s s g ( and so Ž.be bex where x e Ž.ee eBe. Thus eBe End BeBŽ.Be . Now, double centralizer theoryŽ for example,wx CR, Sect. 3D. tells us the following: if A is semisimple, V is an A-module by the representation , s s and C End AŽ.V is the centralizer of A on V, then Ž.A EndC Ž.V .In ( particular, if is faithful, then A EndC Ž.V . In our case, B is semisim- ple and thus the ideal BeB is semisimple, so we only need to show that X BeB acts faithfully on Be. To this end, assume that Žbeb. Be s 0. Then X Žbeb. BeB s 0, but this is the left regular representation of BeB which is X s ( faithful. Thus beb 0 proving that BeB End eBeŽ.Be . UsingŽ.Ž. 5.1.1 c , we have A ( eBe via a ¬ ae. Right multiplication by A makes Be an A-module. Proposition 5.1.3 tells us that ( ( BeB End ABŽ.Be and A End eBŽ.Be . Ž 5.1.4 . Let Xˆ denote the index set for the irreducible representations$ of the algebra X. Double centralizer theory applied toŽ. 5.1.4 gives BeBs Aˆ. Since BeB is an ideal of B, and B is semisimple, there exists an ideal C : B so that B ( BeB [ C Ž.5.1.5 and thus Bˆˆˆs A " C,Ž. 5.1.6 where " denotes a disjoint union. LEMMA 5.1.7. If is a character of BeB, then is completely determined by the ¨alues Ž.ae where a g A. wxg Proof. Žcf. HR, Lemma 2.8. . If beb12 BeB, then from the trace s property of and the map defined inŽ. 5.1.2 , we have Žbeb12 . s s Ž.Ž.ŽŽ...b1 eeb 2 eb21 b e bb21 e . LEMMA 5.1.8. If Q is a set of minimal orthogonal idempotents for A, then eQ s Ä4eq ¬ q g Q is a set of minimal orthogonal idempotents for B. More- o¨er, q is a minimal idempotent of A corresponding to g Aˆ if and only if eq is a minimal idempotent of B corresponding to g Aˆ. Proof. The proof of the first statement usesŽ.Ž. 5.1.1 c and the map XX defined inŽ. 5.1.2 . We have Ž.eq 222s eq s eq, and for q / q g Q, eqeq X s eeqq s 0. If b g B, then Ž.Ž.eq b eq s q Ž ebe . q s q Ž.beqs q Ž.bqes ␣ qqe s ␣Ž.eq , for some scalar ␣,soeq is minimal. 528 TOM HALVERSON Let z be a minimal central idempotent of A such that qz / 0. Since A ( eBe via the isomorphism a ¬ ae, we have qz e / 0. By the fact that A and BeB are full centralizers on Be, their centers coincide, and thus U there is a minimal central idempotent z in BeB which acts by left multiplication on Be the same way that z acts on the right. In particular, U 0 / qz e s qezs zqe, which shows that qe corresponds to g Bˆ. PROPOSITION 5.1.9. If g Bˆ and a g A, then Ž.a , if g Aˆ, s A B Ž.ae ½ 0, if g Bˆˆ_ A. Proof. Žcf.wx HR, Proposition 2.10. . Let P be a complete set of minimal s g orthogonal idempotents of A. Then 1 Ý pg P p. For a A, we have X X X ae s ÝÝpae p s Ýpaep s ÝŽ.Ž.Žep a ep . 5.1.10 . ž/ž/XX X pgPpgPp, p gPp, p gP Using Lemma 5.1.8, extend eP to a complete set Q of minimal orthogo- < nal idempotents of B. For b g B and q g Q, let b q denote the coefficient of q when b is expanded in terms of Q. Let Q denote the elements of Q corresponding to the irreducible component of B indexed by Ždefine P similarly for A.Ž.. Using 5.1.10 , we have s < s X < B Ž.ae Žqaeq .qqqŽ.Ž. ep a ep q . ÝÝÝX qgQqgQ p, p gP Since q, ep g Q, we have ep,ifep s q, qepŽ.s Ž.ep q s ½ 0, otherwise. Moreover, ep g Q if and only if p g P by Lemma 5.1.8, so s < ⌶ . ) A . ⌶ s B иии иии ,Ž. 5.1.11 0.. ⌶ . C ⌶ where C is the character table of C. 5.2. Application to the Partition Algebras There is a natural inclusion : Pxny1Ž.Pxn Ž. Ž.5.2.1 given by adding a horizontal edge at the right end of the partition diagram g s 1 of d Pxny1Ž.. Let ennx E . Then it can be readily checked by drawing 2 s s g diagrams that enne and ea nae nfor all a Pxny1Ž.. Furthermore, Pxy Ž.ª eP Ž. xe n 1 nn n 5.2.2 ¬ Ž. a aen : is an algebra isomorphism. Thus, Pxny1Ž.Pxn Ž.satisfies our properties Ž.5.1.1 of a basic construction. For a partition diagram d, we follow Martinwx Ma4 and define the propagating number, ࠻Ž.d , to be the number of distinct parts of d containing elements from both the top and bottom row of d. Then if d12, d are partition diagrams, we have ࠻ F ࠻ ࠻ Ž.dd12 minŽ. Ž.Ž.d1, d 2 . The ideal PxePxnnnŽ. Ž.is spanned by all the diagrams having a propagat- ing number strictly less than n. Moreover, we havewx Ma3 ( [ ރwx PxnnnnnŽ.PxePx Ž. Ž. S .Ž. 5.2.3 It follows fromŽ. 5.1.5 or directly from the definitionŽ. 1.4.1 that the index set Pˆnnfor the irreducible representations of PrŽ.is a disjoint union s " PˆˆnnP y1 S ˆn .Ž. 5.2.4 g From Lemma 5.1.7 we know that if Pˆny1, then is completely g determined by the values Ž.aennfor a Pxy1 Ž.. Moreover, by Proposi- 530 TOM HALVERSON TABLE I Characters of Px2Ž. л Ž.121 Ž. Ž2 . л xx2 22 Ž.10x 13 Ž.20011 Ž.1002 y11 g g tion 5.1.9, if Pˆnnand a Pxy1Ž., then U Ž.a ,if< < - n, s Pny 1Ž x. P Ž x.Ž.aen U Ž.5.2.5 n ½ 0, if << s n. The character table ⌶ has the form ofŽ. 0.1 . We see this in the PnŽ x. examples in Section 6. 6. CHARACTER TABLES ᎐ s Tables I IV give the characters of PxnŽ.for n 2, 3, 4, 5. The entries in the tables are P Ž x.Ž.d . The rows of the tables are indexed by the n U irreducible representations by the partitions defined inŽ. 1.4.2 , and the columns of the table are indexed by the partitions , which correspond to the standard element d defined inŽ. 2.1.2 . The table for Pxny1 Ž.is embedded in the table for PxnŽ.according toŽ. 0.1 , and so to save space we only give the rightmost part, where << s n, of the tables for n s 4 and s G s n 5. The tables for PrnŽ., r n, are obtained by setting x r. TABLE II Characters of Px3Ž. л Ž.121 Ž. Ž23 . Ž. 3 Ž 2,11 . Ž . л xx322 x 2 x 235 Ž.10xx2 3 x 1410 Ž.200xx 026 Ž.1002 yx 1006 Ž.30000111 Ž.2, 1 0 0 0 0 y102 Ž.1000013 y11 CHARACTERS OF THE PARTITION ALGEBRAS 531 TABLE III Characters of Px4Ž. Ž.43,12 Ž . Ž224 . Ž 2,11 . Ž . л 33 7 715 Ž.1 1 4 5 13 37 Ž.2117931 Ž.12 y11y5531 Ž.3012410 Ž.2, 1 0 y10 220 Ž.1013 y2 y210 Ž.411111 Ž.3, 1 y10y113 Ž.202 y12 02 Ž.2, 12 1 0 y1 y13 Ž.14 y11 1y11 TABLE IV Characters of Px5Ž. Ž.5 Ž 4,1 . Ž 3,2 . Ž 3,122 . Ž 2 ,1 . Ž 2,1 35 . Ž 1 . л 2 4 5 7 12 20 52 Ž.1 1 5 5 13 19 47 151 Ž.2 0 2 3 7 16 42 160 Ž.1002 y1 7 0 32 160 Ž.30133112175 Ž.2, 1 0 0 y2 0 2 22 150 Ž.103 y11 3y9175 Ž.401133715 Ž.3, 1 0 y10 0 1 945 Ž.2002 y1 y32230 Ž.2, 11 0 1 0 0 y3 y345 Ž.104 y11 3y1 y515 Ž.51111111 Ž.4, 1 y10y11 0 24 Ž.3, 2 0 y11y1115 Ž.3, 12 1 0 0 0 y206 Ž2,12 . 0 1 y1 y11y15 Ž.2, 13 y10 1 1 0y24 Ž.115 y1 y11 1y11 532 TOM HALVERSON ACKNOWLEDGMENTS The bulk of this work was done during the Spring 1999 term at the Institute for Advanced Study, whom I thank for their gracious hospitality. I thank Georgia Benkart and Arun Ram for listening to versions of this story and giving advice and Vahe Poladian for implementing the character algorithms. I particularly thank Arun Ram for telling me about the partition algebras and giving me encouragement and numerous suggestions. I also thank the referee for helpful suggestions. REFERENCES wxBr R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. Math 38 Ž.1937 , 854᎐872. wxCP V. Chari and A. Pressley, ‘‘A Guide to Quantum Groups,’’ Cambridge Univ. Press, Cambridge, UK, 1994. wxCR C. Curtis and I. Reiner, ‘‘Methods of Representation Theory: With Applications to Finite Groups and Orders,’’ Vol. I, Wiley, New York, 1981. wxDHP M. Dieng, T. Halverson, and V. Poladian, Frobenius Formulas for the q-rook monoid, preprint, 2000. wxFr F. G. Frobenius, Uber¨ die Charaktere der symmetrischen Gruppe, in ‘‘Sitzungberichte der Koniglich¨ Preussischen Akademie der Wissenschaften zu Berlin, 1900,’’ pp. 516᎐534; Ges. Abhandlungen, Vol. 3, pp. 335᎐348. wxHR T. Halverson and A. Ram, Characters of algebras containing a Jones basic construc- tion: The Temperley᎐Lieb, Okada, Brauer, and Birman᎐Wenzl algebras, Ad¨. Math. 116 Ž.1995 , 263᎐321. wxJo V. F. R. Jones, The Potts model and the symmetric group, in ‘‘Subfactors: Proceed- ings of the Taniguchi Symposium on Operator Algebras, Kyuzeso, 1993,’’ pp. 259᎐267, World Scientific, River Edge, NJ, 1994. wxMac I. G. Macdonald, ‘‘Symmetric Functions and Hall Polynomials,’’ 2nd ed., Oxford Univ. Press, New York, 1995. wxMS P. Martin and H. Saleur, On an algebraic approach to higher-dimensional statistical mechanics, Comm. Math. Phys. 158 Ž.1993 , 155᎐190. wxMa1 P. Martin, Representations of graph Temperley᎐Lieb algebras, Publ. Res. Inst. Math. Sci. 26 Ž.1990 , 485᎐503. wxMa2 P. Martin, ‘‘Potts Models and Related Problems in Statistical Mechanics,’’ World Scientific, Singapore, 1991. wxMa3 P. Martin, Temperley᎐Lieb algebras for nonplanar statistical mechanicsᎏThe parti- tion algebra construction, J. Knot Theory Ramifications 3 Ž.1994 , 51᎐82. wxMa4 P. Martin, The structure of the partition algebras, J. Algebra 183 Ž.1996 , 319᎐358. wxMW1 P. Martin and D. Woodcock, The partition algebras and a new deformation of the Schur algebras, J. Algebra 203 Ž.1998 , 91᎐124. wxMW2 P. Martin and D. Woodcock, On central idempotents in the partition algebra, J. Algebra 217 Ž.1999 , 156᎐169. wxMu1 W. D. Munn, Matrix representations of semigroups, Math. Proc. Cambridge Philos. Soc. 53 Ž.1957 , 5᎐12. wxMu2 W. D. Munn, The characters of the symmetric inverse semigroup, Math. Proc. Cambridge Philos. Soc. 53 Ž.1957 , 13᎐18. wxPu M. Putcha, Monoid Hecke algebras, Trans. Amer. Math. Soc. 349 Ž.1997 , 3517᎐3534. CHARACTERS OF THE PARTITION ALGEBRAS 533 wxRa1 A. Ram, A Frobenius formula for the characters of the Hecke algebras, In¨ent. Math. 106 Ž.1991 , 461᎐488. wxRa2 A. Ram, Characters of Brauer’s centralizer algebras, Pacific J. Math. 169 Ž.1995 , 173᎐200. wxSc1 I. Schur, ‘‘Uber¨ eine Klasse von Matrixen die sich einer gegebenen Matrix zuordnen lassen,’’ Thesis, Berlin, 1901; reprint, in ‘‘Gesammelte Abhandlungen,’’ Vol. I, pp. 1᎐70, Springer-Verlag, Berlin, 1973. wxSc2 I. Schur, Uber¨ die rationalen Darstellungen der allgemeinen linearen Gruppe, 1927; reprint, in ‘‘Gesammelte Abhandlungen,’’ Vol. III, pp. 68᎐85, Springer-Verlag, Berlin, 1973. wxSo1 L. Solomon, The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field, Geom. Dedicata 36 Ž.1990 , 15᎐49. wxSo2 L. Solomon, Representations of the rook monoid, preprint, 1999. wx So3 L. Solomon, Representation of the Iwahori Hecke algebra of MFnqŽ.on tensors, manuscript in preparation. wxWy H. Weyl, ‘‘Classical Groups, Their Invariants and Representations,’’ 2nd ed., Prince- ton Mathematical Series, Vol. 1, Princeton Univ. Press, Princeton, 1946. wxWz H. Wenzl, On the structure of Brauer’s centralizer algebras, Ann. Math. 128 Ž.1988 , 173᎐193.