JOURNAL OF ALGEBRA 203, 91᎐124Ž. 1998 ARTICLE NO. JA977164
The Partition Algebras and a New Deformation of the Schur Algebras
P. MartinU
Department of Mathematics, City Uni¨ersity, Northampton Square, London, EC1V 0H8, View metadata, citation and similar papers at core.ac.ukUnited Kingdom brought to you by CORE
provided by Elsevier - Publisher Connector and
D. Woodcock†
School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London, E1 4NS, United Kingdom
Communicated by Gordon James
Received August 30, 1996
THIS PAPER IS DEDICATED TO THE MEMORY OF AHMED ELGAMAL
P Ž. ޚ The partition algebras Pޚw¨ x n are algebras defined over the ring wx¨ of integral polynomials. They are of current interest because of their role in the transfer matrix formulation of colouring problems and Q-state Potts models in statistical mechanicswx 11 . Let k be a ޚwx¨ -algebra which is a P Ž.P Ž. field, and let Pk n be the algebra obtained from Pޚw¨ x n by base change to k, a finite dimensional k-algebra. If char k s 0 the representation theory of PkŽ.n is generically semisimple, and the exceptional cases are completely understoodwxwx 11, 12, 14 . In 15 we showed that if char k ) 0 and the image of ¨ does not lie in the prime subfield of k, PkŽ.n is Morita equivalent to kᑭ 0 = иии = kᑭ n, a product of group algebras of symmetric groups. In the present paper we show that in the remaining cases in positive
characteristic the representation theory of PkŽ.n can be obtained by localisation from that of a suitably chosen classical Schur algebraŽ. 5.4 .
U E-mail address: [email protected]. † E-mail address: [email protected].
91
0021-8693r98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. 92 MARTIN AND WOODCOCK
The localisation functor is directly analogous to the Schur functor, and provides a powerful tool for the transfer of information from the Schur algebras to the partition algebras. In particular we express the decomposi- tion matrices of the partition algebras as submatrices of those of the Schur algebrasŽ. 5.7 . Our approach also allows us to recover the exceptional structure of the partition algebra in characteristic zero by passing to large positive characteristic and applying the Jantzen sum formulaŽ. 6.4 . At the heart of our analysis is a new algebra Tk which forms a bridge between the representation theory of the partition algebras and of the general linear groups. The precise definition of the algebra Tk is rather technical, but roughly speaking it is a simultaneous deformation of all the classical Schur algebras, obtained by letting the lengths of the first rows of the relevant Young diagrams tend to infinity. After some preliminary categorical remarks in Section 1, we introduce in Section 2 a global version of the partition algebra. This is the quotient by certain ‘‘chromatic relations’’ of an auxiliary algebra defined in terms of the pasting together of finite graphs. We define the Potts modules, and show thatŽ. in a suitably qualified sense the partition algebras can be viewed as ‘‘generic limits’’ of the endomorphism algebras of the Potts modules, considered as modules for the symmetric groups. Motivated by this observation, we recall in Section 3 the definition of the Schur algebras as endomorphism rings of symmetric group actions on tensor space, and define our new algebra Tk as a ‘‘generic limit’’ of Schur algebras. The remaining three sections are devoted to the representation theory of Tk and of the partition algebras. In Section 4 we examine some aspects of the representation theory of Tk which do not depend on the characteris- tic of the base field. We parameterise its simple modules in terms of their highest weights, and show using Green’s codeterminantswx 8 that Tk is close to being quasi-hereditary. In Section 5 we work in positive character- istic, showing that arbitrarily large finite pieces of the representation theory of Tk can be interpreted in terms of the Schur algebras. In this way we ‘‘reduce’’ problems in the modular representation theory of the parti- tion algebras to problems in the representation theory of the general linear groups. In the final section we show that in characteristic zero Tk is Morita equivalent to the global partition algebra, and hence that it is generically semisimple. We analyse the exceptional cases by reduction to positive characteristic.
1. PRELIMINARIES
Let A be a ring not necessarily with identity. We write A-Mod for the category of A-modules V satisfying AV s V; representation-theoretic PARTITION ALGEBRAS 93 properties of A, such as Morita equivalence and semisimplicity, are to be understood with respect to this category. Let E be a set of pairwise orthogonal idempotents in A. We say that E is complete if A s Ýe, f g E eAf. In this case an A-module V belongs to A-Mod if and only if V eV. s [eg E Such rings arise naturally as follows. Let X ( " iiX be a fixed decom- fin position of an object in an abelian category C, and write End C Ž.X for fin "i, j Hom C Ž.Xij, X . We refer to the elements of End CŽ.X as the finitary endomorphisms of X. The projections onto the summands Xi form a fin complete set of pairwise orthogonal idempotents in End C Ž.X . Although we do not emphasize the fact notationally, this definition depends very much on the choice of decomposition. We remark however that if C is a Grothendieck category, and if the summands Xi have finite fin length, End C Ž.X is at least unique up to isomorphism. For we may refine the given decomposition to one by indecomposable objects without chang- ing the finitary endomorphism ring. These objects have local endomor- phism rings by Fitting’s lemma, so Azumaya’s theorem shows that the decomposition is unique up to isomorphism. We will make frequent use of the language of localisationŽ see Gabriel wx6, Sect. III. , and for the convenience of the reader we recall the facts we will need here. A strict full subcategory L of a Grothendieck category C is a localising subcategory ifŽ. i given any short exact sequence 0 ª X ª Y ªZª0inC we have Y g L if and only if X, Z g L ;iiŽ.Lis closed to taking arbitrary colimits. In this case there is a quotient category CrL and a localisation functor F: C ª CrL , universal among exact functors which annihilate L. Moreover F has a right adjoint G, the section functor. The simple objects in CrL are the images of the simple objects in C _ L. If E is any set of pairwise orthogonal idempotents in a ring A, the functor A-Mod ª EAE-Mod V ¬ EV identifies EAE-Mod with the localisation of A-Mod at the subcategory consisting of those V with EV s 0. This is a mild generalisation of Green wx7, Sect. 6 . We will abuse terminology and refer to AEA as a localisation of A. A Grothendieck category C is said to be locally finite if every object of C is the union of its finite length subobjects. In this case, if V, L g C with Lsimple, we write wxV: L for the composition multiplicity of L in V, which is defined to be
wxV:LssupÄ4 wW : L xN W F V, W of finite length .
We write ގ Ž.resp. ގ0 for the natural numbers withoutŽ. resp. with zero. 94 MARTIN AND WOODCOCK
2. PARTITION ALGEBRAS AND POTTS MODULES Ž.2.1 The Graph Algebra. Let g be a finite graph. We will usually confuse g notationally with its vertex set, leaving its edge set to be understood from the context. Write n Žresp. nX . for the setÄ4 1, . . . , n Žresp. Ä1,...,XXn4.Ž..Byan n,m-graph we mean a finite graph g together with a map
X g : n j m ª g.
X Thus g labels the appropriate vertices with the elements of n j m .A given vertex may have many labelsŽ. or none at all . We refer to the vertices labelled by n Žresp. mX .Ž.as the inputs resp. outputs of g, and call them collectively the exterior ¨ertices of g. If g is an Ž.n, m -graph we write ;g for the equivalence relation on X njm given by i ;ggj if and only if Ž.i and g Žj .lie in the same connected component. If h is a second Ž.n, m -graph we write g « h to X X mean that ;ghis contained in ; ŽŽas subsets of n j m .Ž= n j m ... A morphism of Ž.n, m -graphs is a graph morphism which commutes with the structure maps. We write GŽ.n, m for the set of isomorphism classes of Ž.n,m-graphs. If n s m we abbreviate this to G Ž.n . We do not distinguish notationally between a graph and its isomorphism class.
Let k be a commutative ring. The graph algebra Gk is the free k-module with basis D n, m GŽ.n, m . The product gh g GŽ.l, n of g g G Ž.l, m and hgGŽ.m,nis obtained by identifying the outputs of g with the inputs of X h. Thus gh agrees with g on l and with h on n . We declare all other products of basis elements to be zero, and extend bilinearly to an associa- tive product Gkk= G ª G k. One should compare this construction with the tangle categorywx 4 . Ž.2.2 The Partition Algebra. We say that a connected component of an Ž.n,m-graph g is isolated if it contains no exterior vertices, and that a vertex x is isolated if Ä4x is itself an isolated connected component of g. Suppose now that k is a ޚwx¨ -algebra, and let IŽ.n, m be the submodule of k GŽ.n, m generated by elements of the form
gj˙ Ä4isolated vertex y ¨g,1Ž. Ž.Ž.gjqÄ4ⅷⅷ gjy Ä4ⅷ g,2Ž. for g g GŽ.Ž.n, m . In 2 the first term is g with an extra edge inserted between two of its vertices, while the second is g with those two vertices PARTITION ALGEBRAS 95 identified. For example, we have the following congruences modulo IŽ.1, 1 :
We callŽ. 1 and Ž. 2 the chromatic relations because of their close connec- tion with chromatic polynomialsᎏseeŽ. 2.3 . The sum I of the I Žn, m .is evidently an ideal in Gkk, and we write P for the corresponding quotient, which we call the global partition algebra. Write : Gkkª P for the canonical surjection. We say that e g GŽ.n, m is a partition if it has no edges and no isolated vertices. We write EŽ.n, m for the set of these and abbreviate E Ž.n, n to EŽ.n. The map e ¬;e restricts to a bijection from E Žn, m .onto the set X of equivalence relations on n j m . Write PkkŽ.Žn, m resp. P Ž..n for the image under of k GŽn, m . Žresp. k G Žn ... From Ž 1 . and Ž 2 . we see Ž using, for example, a quick application of Bergman’s diamond lemmawx 2. that the image of EŽ.n, m is a k-basis of Pk Ž.n, m . We thus abuse notation and write EŽ.n, m for this image too. It is easy to check that the algebra PkŽ.n is canonically isomorphic to the partition algebra over k as defined inwx 15 .
Ž.2.3 Chromatic Polynomials. For e g E Žn, m .define a map De: GŽ.ޚ P Ž. Gn,mª wx¨ by writing in Pޚw¨ x n
Ž.g s Ý DgeŽ.иe.3Ž. egEŽ.n,m
One checks easily using the chromatic relations that De is also given by
Ž.h ␥Žh.y␥Že. DgeŽ.syÝ Ž1 . ¨ , h where the sum ranges over all bond subgraphs h of g such that h m e; Ž.h and ␥ Ž.h denote respectively the numbers of edges and connected 96 MARTIN AND WOODCOCK components of h.Ž A bond subgraph of g is one obtained by deleting a subset of the edges of g..
If n s m s 0, Dл is the usual chromatic polynomial wx16 . Our De should thus be thought of as a kind of relative chromatic polynomial.
Ž.2.4 The Potts Module. Let k be a commutative ring. Fix Q g ގ and consider k as a ޚwx¨ -algebra via ¨ ¬ Q. For each such choice of ޚ wx¨ -alge- bra structure on k we define a right Gkk-module U as follows. As a k-module
Ukks UnŽ.,4Ž. n"gގ
Qmn where UnkŽ.s Žk. . The latter module has a k-basis Ä4¨i for i ranging over the set of maps n ª Q. For i: n ª Q and j: m ª Q, let Ti, jk: UnŽ.ªUm k Ž .be the k-map defined by ¨ liT ,jis ␦ ,lj¨ . Extend this to a map Ti,jk: U ª U kby declaring X it to be zero on all other summands. Write i j j: n j m ª Q for the function which is i on n and j on mX Ž.after stripping off the primes . If l is any function, let ²:l denote the partition of its domain by the fibres of l. Define : Gkkkª End Ž.U by
X Ž.gsÝD²ijj: Ž.Ž.gQTi,j, ggGŽn,m .,5 Ž. i,j where
X Defs ÝD , e g E Ž.n, m . f«e
Note that any g g GŽ.n, m kills Uikk Ž.except for i s n, and maps UnŽ. into UmkŽ.. It is straightforward to check that is a morphism of k-algebras. Moreover it follows from the chromatic relations that factors through : k Gkkª P ,soU kbecomes a Pk-module, which we call the Potts module. The PkkŽ.n -submodule UnŽ.of Ukis the usual Potts module wx13 . FromŽ. 3 and Ž. 5 we see that on elements of E Žn, m .we have
Ž.e s Ý Ti,j.6Ž. e«²:ijj
Ž.2.5 Remark. There is a slightly more general version of the above, where one works over ޚwxu, ¨ instead of ޚ wx¨ , and replaces the second chromatic relationŽ. 2 by
Ž.Ž.gjyÄ4ⅷⅷ ugjy Ä4ⅷ g. PARTITION ALGEBRAS 97
The polynomials De then become relative versions of the dichromatic polynomials wx1 . InŽ. 2.4 one makes k into a ޚwxu, ¨ -algebra via ¨ ¬ Q, u ¬ R, where R is some fixed element of k. Take k s ރ and R s expŽ. 1rT y 1 ŽT g ރ _ Ä40. . Evaluating the Ž.ordinary dichromatic polynomial Dgл Ž.at u s R and ¨ s Q gives the partition function ZŽ. g of the Q-state Potts model on g with temperature T
Ž.in appropriate unitswx 1 . Let bnnŽ.resp. d be the canonical discrete Ž. 0, n ŽŽ..Žresp. n, 0 -graph. Here b and d stand for ‘‘birth’’ and ‘‘death.’’. For an
Ž.n,m-graph g, bgdnmis the underlying graph of g. It is easily checked that
ZgŽ.s Žbnm .Ž.Žgd ..7Ž.
Let An denote the undirected graph with verticesÄ4 0, 1, . . . , n y 1 and edges ÄÄi, i q 1: 4 is0, 1, . . . , n y 2.4 In d-dimensional physics there is particular interest in hyper-cubical lattices, i.e., graphs of the form A n1 = иии = A , considered either as crystal lattices or as discrete approxima- n d d tions to ޒ .Ifhis a graph with vertices h1,...,hrn, we regard h = A as an Ž.r, r -graph with inputs Ž.Ž.h1,0 ,..., hr, 0 and outputs Ž.h1, n y 1, ...,Ž.hrn,ny1 , and h = A __ h = A 1as the same Ž.r, r -graph except for having the edges of h = A1 removed. Clearly
n Ž.h = Anq11__ h = A s Ž.Ž.h = A 21__ h = A ,
soŽ. 7 yields an iterative technique for computing the partition function and corresponding physical observables of the Potts model on such lattices.
In this setting the matrix Ž.h = A21__ h = A is called the transfer matrix, and physical attention is focused on its eigenvalue spectrumwx 1 .
Ž.2.6 The symmetric group ᑭ Qkacts from the left on U by ‘‘permuting colours,’’ i.e., if i: n ª Q and g ᑭ Qi, ¨ s ¨ik. The action of G is clearly invariant under change of colour, so the actions of the symmetric
group and of the partition algebra on Uk centralise one another. We can be more precise.
Take T s Ýi, jiaT,ji,jkkkgHom ŽŽ.Ž..Un,Um. Then T is a kᑭ Q-map if and only if a i, jis a ,jQfor all g ᑭ and all i, j: n ª Q, i.e., if and only if a is constant on the orbits of the ᑭ Q-action on the set of maps n mX Q. The finitary endomorphism ring Endfin Ž.U with respect to j ª k ᑭ Q k the decompositionŽ. 4 thus has a basis ÄŽ.e 4, where
Ž.e s Ý Ti,j,8Ž. es²:ijj 98 MARTIN AND WOODCOCK
X and e ranges over the partitions of all n j m with at most Q parts, for n,mG0. We can now read off the following resultŽ cf. Joneswx 10 , Martin wx13. :
Ž.2.7 LEMMA. Ž.i ŽP . Endfin Ž.U . kks ᑭQk Ž.ii If Q G n q m, induces an isomorphism
P n, m Hom Un,Um. kkŽ.( ᑭQŽ.kk Ž.Ž.
Proof. This follows immediately from the preceding remarks and Mobiøus¨ inversion applied toŽ. 6 . Ž.2.8 Remarks.
Ž.i We view this lemma as saying that Pk is the ‘‘generic limit’’ of the endomorphism algebras of the Potts modules as Q ª ϱ. Ž.ii If k is a field of characteristic zero or characteristic p ) Q, kᑭQ is semisimple by Maschke’s theorem. Thus Ž.Pk is semisimple, being the endomorphism algebra of a semisimple module, and if Q G 2n the partition algebra PkŽ.n itself is semisimple. Ž. Ž. Put Pk F n s Ý0 F i, jF nkP i, j , another localisation of P k.
Ž.2.9 LEMMA. Suppose k is a field and take n G 1. Then PkŽ.F nis Morita equi¨alent to PkŽ.n if and only if ¨ is non-zero in k.
Proof. Let eikkbe the identity element of P Ž.i , an idempotent in P .
Suppose that ¨ is non-zero in k. We must show that e0 ,...,eny1 belong to the ideal in PknŽ.F n generated by e . Define an element e i,jg EŽ.i, j by
XXXXX Ä4Ä41,1 ,...,Ä4iy1, Ž.i y 1,Äi,i, Ž.iq1 ,..., j 4if i F j ei,js XXX ½Ä4Ä41,1 ,...,Ä4jy1, Ž.j y 1,Äj,jq1,...,i, j 4 if i G j.
X ŽŽHere we write elements of E i, j.as partitions of i j j ..Ž. By 2 , multipli- cation in Pk gives
ei if n G i G 1 eeei,nnn,is ½¨e0 if i s 0, as required.
Suppose conversely that PkkŽ.n and P ŽF n .are Morita equivalent. Since k is a field these algebras are artinian, hence have the same finite number of simple modules. The localisation V ¬ eVn therefore kills no PARTITION ALGEBRAS 99 simple modules, hence is an equivalence. The natural map PknŽ.F ne eV V is thus an isomorphism for all V P Ž.n -Mod, in mPk Žn. n ª g k F particular PknkkŽ.F ne P Ž.Fn sP Ž.Fn. However, it is easy to see that e0 belongs to the ideal generated by en if and only if ¨ is non-zero in k, and we are done.
3. SCHUR ALGEBRAS
In this section we introduce a new algebra defined over the ring of numerical polynomials which serves as a bridge between the representa- tion theory of the partition algebras and that of the general linear groups. Ž.3.1 Global Schur Algebras. We begin by recasting some basic results on the classical Schur algebras from Greenwx 7 in a slightly different form. Put
⌳ s Ä4: ގ ª ގ0N suppŽ. is finite
⌳000sÄ4:ގªގNsuppŽ. is finite .
The elements of ⌳ or ⌳ 0 are called weights.If is a weight we usually write iiinstead of Ž.i . The numbers are called the parts of . The degree of is << s Ýii ; it is finite by definition. If ⌫ is any set of weights and Q g ޒ we write ⌫Ž.ŽQ resp. ⌫wxQ .for the subset of ⌫ consisting of those weights of degree Q Ž.resp. of degree at most Q . The dominant weights are
q ⌳sÄ4g⌳N12GGиии q ⌳0001sÄ4g⌳NGGиии .
We define the dominance order G on ⌳ 0 in the usual manner, i.e.,
jj
Gif and only if ÝÝijG ᭙j g ގ0. is0is0 We define the dominance order on ⌳ by
G if and only if Ž.Ž.<< , 12, ,... G <<,1, 2,... , where the order on the right is that already defined on ⌳ 0.If <<s < Ž.Qy<<,12,,... GŽ.Qy < <,1, 2,... in ⌳ 0. For any set X put IŽ.ގ00, X s Ä4i: X ª ގ N suppŽ.i is finite . We call the elements of IŽ.ގ0 , X indices, and as with weights we write i x for the value of the index i at x g X.IfQgގwe write IŽ.ގ0 , Q for IŽ.ގ00,Q.If igI Žގ ,X .we define wt Ž.i g ⌳ by wtŽ.i js Ä4x g X N i xs j . If X is finite we also define wt 00Ž.i g ⌳ by the same formula; thus in this case wtŽ.i is the restriction of wt 0Ž.i to ގ. Let k be a commutative ring. Put E ke , the free k-module on kis [iG0 mQ the countable set Ä4e01, e , . . . . Take Q g ގ. The Q-fold tensor power Ek has k-basis Äe i IŽ.ގ , Q 4, where e e иии e . For ⌳ Ž.Q i N g 0 iis 1m m iQ g 0 put MkiŽ. s keÄ4Nwt 0 Ž.i s . The symmetric group ᑭ Q acts from the right by place permutation on mQ EkQ, and we have a decomposition of kᑭ -modules EmQMŽ..9Ž. kks[ g⌳0Ž.Q The MkŽ. are permutation modules associated to Young subgroups of ᑭ . Put SQŽ. Endfin Ž EmQ ., the ring of finitary endomorphisms with Qks kᑭQk respect to the decompositionŽ. 9 , which we call the global Schur algebra of degree Q.Itisa k-algebra which is free of countable rank over k. Let gSQkŽ.be the idempotent projecting onto the summand MkŽ. , and if n g ގ let be the sum of all for of degree Q with i s 0 for i G n. Then SnkkŽ.,Q[SQ Ž.is the usual Schur algebra, and it follows from PARTITION ALGEBRAS 101 wx7, 6.5 that if n G Q this localisation defines a Morita equivalence of SQkkŽ.with Sn Ž,Q .. For i, j g I Žގ0, Q .let i,jkg SQ Ž.be the map which sends em to ÝŽi, j.; Žl, m. el , where ; denotes conjugacy under the right 2 2 ᑭQ-action on IŽ.ގ0 , Q . Bywx 7, Sect. 2 , Äi, j N Ž.i, j g I Žގ0 , Q .r;4is a k-basis of SQkŽ.. Note that if wt 0Ž.i s we have i, i s . Multiplication of basis elements is given by the formula i,jl,mpsÝZiŽ.,j,l,m,p,q,q,1 Ž.0 Ž.p,q 2 where the sum is taken over a transversal ÄŽ.p, q 4of IŽ.ގ0 , Q r; , and ZiŽ.Ž.,j,l,m,p,q s࠻Ä4sgIގ0,QiŽ.Ž.,j;p,s,Ž.Ž.s,q;l,m. Ž.11 ŽSee Greenw 7,Ž. 2.3bx .. Ž.3.2 DEFINITION. Put A Ä4g ޑ gŽ.ޚ ޚ ޚ¨ , s g wx¨ : s[ž/i iG0 the ring of ‘‘numerical polynomials.’’ There is a natural action of ޚ on A by translation: if g g A and i g ޚ define giwxgAby gi wxŽ. j sgi Žqj .. For N g ގ we write N A for the subring of A generated by ¨¨, , . . . ¨ . It follows easily from the relation ž/ž/01 ž/N ¨¨y1¨y1 sq Ž.jgގ ž/ž/ž/jjjy1 that N A is closed under the translation action of ޚ. ŽQ. If g ⌳ and Q g ގ with Q G << define g ⌳ 0Ž.Q by ŽQ. Nsގ ŽQ. 0 sQy<<, and for ⌫ : ⌳ put ŽQ.ŽQ. ⌫ sÄ4 Ng⌫,<<FQ. ŽQ. If i g IŽ.ގ0 , ގ has support in Q, we denote by i its restriction to Q. ŽQ. ŽQ. Then wt 0Ži .Ž.s wt i . Let ᑭϱ be the group of permutations of ގ. For i, j, l, m g IŽ.ގ0 , ގ write Ž.Ži, j ; l, m .if the pairs are conjugate under the right action of ᑭϱ 102 MARTIN AND WOODCOCK 2 on IŽ.ގ0 , ގ . Take symbols Äi, j N i, j g IŽ.ގ0 , ގ 4with i, jls ,mif and only if Ž.Ži, j ; l, m .. Let TA be the free A-module with basis Äi, j N Ž.i, j 2 gIŽ.ގ0,ގr;4. The proof of the following proposition will occupy us for the next few subsections. Ž.3.3 PROPOSITION. There are unique elements ZˆŽ. i, j, l, m, p, q g A such that Ž.i If the supports of i, j, l, m, p, q are contained in Q then ŽQ.ŽQ.ŽQ.ŽQ.ŽQ.ŽQ. ZiˆŽ.,j,l,m,p,qQŽ.ŽsZi ,j ,l ,m ,p ,q .,with Z as in Ž.11 . Ž.ii If all the parts of the weights of i, j, l, m, p, q are less than or equal to some N g ގ, then ZˆŽ. i, j, l, m, p, q lies in N A. Ž.iii The definition ˆ i,jl,mpsÝZiŽ.,j,l,m,p,q,q,1 Ž.2 Ž.p,q 2 where the sum is taken o¨er a trans¨ersalÄŽ. p, qofI4 Ž.ގ0 ,ގr;,makes TA into an associati¨e A-algebraŽ. without identity . Ž.3.4 Remarks. Ž. i If the weights of i, j, l, m have degrees bounded by Qr2, one can replace them by new elements having support in Q without changing the elements i, jland ,m. The summation inŽ. 12 can also be restricted to pairs p, q with support in Q, so the sum inŽ. 12 is finite. We see that under the specialisation A ª ޚ, ¨ ¬ Q we recover a part of the multiplication of SQޚŽ.. This remark is formalised in Ž 3.9 . below. Ž.ii The uniqueness statement is clear fromŽ 3.3 .Ž. i , since this speci- fies the value of Zˆ at infinitely many integers Q. Moreover, once we have proved the existence of numerical polynomials satisfyingŽ.Ž. 3.3 i , part Ž.Ž.3.3 iii is immediate: associativity amounts to checking certain identities among numerical polynomials, which we know hold at infinitely many values of Q by the associativity of multiplication in the Schur algebras. It is thus enough to prove that the function of Q defined for large Q byŽ.Ž. 3.3 i is indeed a numerical polynomial, and that it satisfies the condition Ž.Ž.3.3 ii . Ž.iii If k is an A-algebra we write Tk for the k-algebra TAmA k.As with the Schur algebras, we write for l, l if l has weight . The form a complete set of pairwise orthogonal idempotents in Tk . Ž.3.5 EXAMPLE. To display a basis element i, j we write the values of the maps i and j as lists extending to the right, with i above j and trailing PARTITION ALGEBRAS 103 zeroes omitted. As a sample multiplication the reader can check that 11000 11111 211 3 110 4 1100 ¨ y ¨y ¨ y . 11111 11000sqž/ 3 11 ž/ 2 101 q ž/ 1 0011 Ž.3.6 The proof ofŽ. 3.3 is purely combinatorial, and we approach it by formulating a slightly more general problem. Temporarily we will allow weights taking negative entriesŽ. for convenience only . We say that a weight in this new sense is non-negati¨e if all its entries are so, i.e., if is a weight in the old sense. ⌽ ŽŽ..⌽ Let A be a finite set. Given a collection s a ag A of not neces- ŽŽ.. sarily disjoint finite sets, and a family s a ag A of weights put nŽ.⌽, ࠻ f : ⌽ Ž.a ގ wt Ž.f ⌽ Ž.a᭙aA. s ½5Dª 00NsŽa. g agA This is evidently zero unless all Ž.a are non-negative and <⌽Ž.a < ␦,r ⌽Ž.a j r if ␦ Ž.a s 1 ⌽Ž.a s ½⌽Ž.a if ␦ Ž.a s 0 ␦,r Ž.a s Ž.aq␦ Ž.Žar,0,0,... . . ␦,r ␦,r Ž.3.7 LEMMA. For fixed Ž.⌽, and ␦ as abo¨e, nŽ⌽ , .is a numerical polynomial in r. More precisely, if Ž.a i F N for all a g A and all ␦, r ␦ , r i ) 0, then nŽ⌽ , . is gi¨en by an element of N A ␦,r ␦,r Proof. If ␦Ž.a s 0 for all a g A, nŽ⌽ , .is independent of r and we are done. Thus suppose that ␦Ž.b s 1 for some b g A and put X X X A s A j Äb 4 where b is some new element not appearing in A. For fixed ⌿ ŽŽ..⌿ X rdefine s a ag A by ␦ , r ¡⌽Ž.a if a g A _ Ä4b ⌿Ž.as~⌽Ž.b if a s b X ¢r if a s b , i.e., we break each set ⌽Ž.b ␦,r into two pieces: an r-independent piece X ⌽Ž.band an r-dependent piece ⌽ Žb .s r. 104 MARTIN AND WOODCOCK Let ⌬ be theŽ. infinite set of weights with << s 0 and 12, ,...G0. Ž Ž.. X For such define s a ag A by ␦ , r ¡Ž.a if a g A _ Ä4b Ž.as~Ž.by if a s b X ¢qŽ.r,0,0,... if asb. ␦,r The g ⌬ parameterise all the ways to break up the weight Ž.b into two parts, one for ⌽Ž.b and one for r. Note that since << s 0, adding to a weight does not change the degree of , although it will often happen that the new weight q has negative partsŽ and hence contributes nothing to our sums. . We now have ␦ , r ␦ , r nŽ.Ž.⌽ , s Ýn ⌿, .1Ž.3 g⌬ The contribution due to a given g ⌬ is zero unless Ž. i ÝjG1 jFr. Ž.ii iiF Ž.b for all i ) 0. In particularŽ. 13 is a finite sum. We claim that for all g ⌬ r r y 112r y y n⌿,nŽ.⌽,␦ иии ,14Ž. Ž.sy ž/1ž/ž/23 ␦ ŽŽ. ␦ Ž.. where y is shorthand for a y a ag A. This is enough, since the first factor is independent of r, while the binomial product is a numerical polynomial in r lying in the appropriate subring of A byŽ. ii and Ž.3.2 . Ž. Consider the maps f counted by the left hand side of 14 . If Ý jG1 j ) r there are no such maps byŽ. i , and the binomial product is zero. If Ž. ÝjG1jFr, the number of maps f satisfying the constraint wt 0f Nsr Ž. r ry1 qr, 0, . . . is precisely иии . For each such choice of f N r there ž/1 ž/2 are nŽ.⌽, y ␦ choices for the rest of f. Completion of the Proof of Ž.3.3 . Fix R g ގ so that the supports of i, j, l, m, p, q lie in R, and take Q G R. To show that the coefficient ŽQ.ŽQ.ŽQ.ŽQ.ŽQ.ŽQ. ZsZiŽ ,j ,l ,m ,p ,q .is a numerical polynomial in Q we may assume by weight considerations that j s l, i s p, and q s m for some g ᑭ R. By definition, the value of Z at Q is the number of indices sgIŽ.ގ0 ,Qsuch that ŽiŽQ.Ž,jQ..Ž;iŽQ.,s. Ž.Ž.jŽ.Q,qŽ.Q ;Ž.s,qŽQ.. PARTITION ALGEBRAS 105 For each b g ގ0 let ⌽Ž.i, b be the set of c g R such that ic s b, and let Ž.i,bbe the weight of j restricted to ⌽Ž.i, b . Define ⌽ Žq, b .analogously, but with i replaced by q, and let Ž.q, b be the weight of j restricted to ⌽Ž.q,b. Put A s ÄŽ.Ž.i, b , q, b N b g ގ04and ␦Ž.i, b s ␦ Žq, b .s ␦b,0 Žb ␦,QyR␦,QyR gގ0 .Ž. The value of Z at Q is then n ⌽ , .Ž.as in 3.6 . Thus byŽ. 3.7 we see that Z is a numerical polynomial in Q y R of the appropriate type, hence also in Q. Ž.3.8 DEFINITION. Let V be a module for Tkkor SQ Ž.. We adopt the usual terminology of weight spaces, calling V the -weight space of V, and writing it simply V. Generalising slightly, if ⌫ is a subset of the ⌫ appropriate weight set we put V s Ýg ⌫ V. If the weight spaces of V are finite dimensional, we define itsŽ. formal character ch Ž.V by chŽ.V s ÝŽ.dim Ve, where the e are formal exponentials. We use analogous notions and notation for right modules. We write TkkŽ.⌫ and S Ž⌫, Q .for the localisa- ⌫⌫⌫⌫ tions Tkkand SQŽ., respectively. Evaluation at any Q g ޚ gives a ring homomorphism A ª ޚ, and if k is any commutative ring we write kŽQ. for k made into an A-algebra via this map. We show now that when Q g ގ there is an isomorphism between finite pieces of TkkŽQ.and SQŽ.. Ž.3.9 LEMMA. The specialisation f ¬ fŽ. Q induces an isomorphism of rings TkkŽQ.Ž.⌳wxQr2(SŽ.⌫,Q, where ŽQ. ⌫sÄ4Ng⌳wxQr2sÄ4g⌳0012Ž.QNGqqиии . Proof. The algebra Tk ŽQ.Ž⌳wxQr2. is the k-span of those basis elements i, j with i, j having weights in ⌳wxQr2 . Under this restriction there are at most Q elements a g ގ with Ž.Ž.i aa, j / 0, 0 . We can thus replace i and j by indices with support in Q without changing i, j. With this choice of representing indices, mapping i, jito ŽQ.Ž,jQ.defines a k-linear isomor- phism TkkŽQ.Ž⌳wxQr2 .Ž.ª S ⌫, Q . The multiplication in TkŽQ.is defined precisely to make this map an isomorphism of algebras. Ž.3.10 We show now how to recover the global partition algebra as a localisation of TA. The essential idea is to identify the Potts module Uk ŽŽ..see 2.4 , viewed now as a right ᑭ Q-module, with a summand of the mQ defining module Ekkof SQŽ.Žsee Ž 3.1 .. , and then to check that this 106 MARTIN AND WOODCOCK identification is compatible with ‘‘taking generic limits.’’ The identification we use is non-canonical as it depends on the choice of the function to be defined below. Put EsÄ4eNeis a partition of n for some n g ގ0 . Thus each e g E is a set of disjoint subsets of ގ with D e s n for some n G 0. Fix a function with the following properties: Ž.i The domain of is E. Ž.ii For each e g E, Ž.e is an injection e ª ގ. Ž.iii im Ž.e s im Ž.f if and only if e s f. For e g E define Ž.e g ⌳ by 1ifigim Ž.e Ž.eis ½0 otherwise. ByŽ. iii we have Ž.e s Ž.f if and only if e s f. Let H ; ⌳ be the set of all weights Ž.e so obtained. For each e g E, the function Ž.e gives a labelling of the parts of e by distinct positive integers, in such a way that no two distinct partitions have the same set of labels. The weight Ž.e records which of the labels are used for e. For i: n ª Q define Ž.i : Q ª ގ0 by 1 Ž.²:iiŽ.y Ž. j if j g imŽ.i Ž.ijs Ž.15 ½0 otherwise. It is routine to verify that Ž.iv is an ᑭ Q-equivariant injection. Ž.vwt00 ŽŽ..iswt ŽŽ.. j if and only if ²:i s ²:j . Ž²:Recall that i denotes the partition of n given by the fibres of the map i..Ž Here we have shifted the natural left action of ᑭ Q on IQ,n.to the right by inversion. Since all the algebras involved are isomorphic to their opposites, the choice of left or right is purely one of taste. Note that by Ž.iv , defines a bijection ; ŽQ. IQŽ.,nªÄ4igI Žގ0,Q .Ž.wt i g H . The maps i: n ª Q for n g ގ0 parameterise the basis Ä4¨i of the Potts mQ module, and extending the map ¨i ¬ eŽi. additively gives a map Uޚޚª E , PARTITION ALGEBRAS 107 inducing an isomorphism of right ޚᑭ Q-modules Uޚޚ( M Ž. . "ŽQ. gH Each partition e of n into at most Q parts determines the permutation module summand ޚ of UnŽ., which corresponds underŽ. 16 to [²i:se ¨i ޚ MޚŽŽ..e. We thus have an induced isomorphism of finitary endomorphism rings: Endfin U SHŽQ.,Q.16 ޚᑭQŽ.ޚޚ( Ž. Ž. Recall fromŽ. 2.6 that the left hand side has a basis consisting of all X Ž²i j j :., for i: n ª Q, j: m ª Q, m, n G 0. This element corresponds underŽ. 16 to Ži., Ž j., so fromŽ. 6 and Ž 2.7 . we have a surjection of rings ŽQ. PޚޚŽQ. ª SHŽ.,Q e¬ÝŽi.,Žj..1Ž.7 e«²:ijj Here if e g EŽ.n, m , the sum is restricted to those maps i: n ª Q and j: X mªQ. This map is injective on restriction to PޚŽQ.Ž.n, m if Q G n q m byŽ. 2.7 again. Ž.3.11 The set Hqof dominant conjugates of elements of H is a Hqs Ä4Ž.1 a G 0. We call weights of this form hook weights since ŽQ. is a hook weight in the usual sense for Q G << . PROPOSITION. Let k be an A-algebra. Ž.i The k-algebras Pkkand T Ž.H are isomorphic; thus Pkis Morita q equi¨alent to the localisation of Tk at H . Ž.ii If n G 1, PkkŽ.Žn resp. P ŽF n .. is Morita equi¨alent to the q q localisation of Tkat Hwx n _ ÄŽ.0 4Žresp. Hnwx.. Proof. Ž.i For i: n ª Q define Ž.i : ގ ª ގ0 byŽ. 15 , and define an A-linear map : PA ª TA Ž.H by the formulaŽ.Ž 17 using the new .. This is an isomorphism of A-modules which byŽ. 3.9 induces the injection PޚޚŽQ.¨SHŽ.,Qconstructed above whenever Q G 2 maxÄ4n, m . Thus is itself an algebra isomorphism, which induces the required isomorphism PkkªTŽ.Hby base change. 108 MARTIN AND WOODCOCK Ž.ii This follows because the set of dominant conjugates of weights of the form Ž.e for e a partition of m is Hmqwx_ÄŽ.0if4Äm)0 andŽ. 0 4 if m s 0. 4. CHARACTERISTIC-FREE REPRESENTATION THEORY Ž.4.1 Tableaux. We recall some ideas from Greenwx 7, Sect. 4 . The Young diagram of a weight g ⌳ 00is the subset wx of ގ = ގ defined by wxsÄ4Ž.a,bgގ0=ގNbFa.1Ž.8 We adopt the matrix convention for drawing Young diagrams, so that, for 2 example, if s Ž5, 4 , 1. we have IIIII IIII wxs IIII I Define a -tableau to be a map T: wx ª ގ0 , and declare T to be semi-standard if the values of T are non-decreasing along the rows of wx , and standard if it is semi-standard and also the values are strictly increas- ing down the columns of wx . We fix once and for all a -tableau T which is bijective onto Q, where Q s << , and call it the basic -tableau. For igIŽ.ގ0 ,Qits -tableau Ti is defined to be the composite iT : wx ª Q ªގ0. We declare i to beŽ. semi- standard if Ti is so. This depends upon the choice of T . Any -tableau T can be written T s Ti for a unique choice of i g IŽ.ގ00, Q , and we put wt Ž.T s wt 0 Ž.i and wt Ž.T s wt Ž.i . The canonical index lŽ. of weight is defined by requiring its -tableau to be 00000 иии 1111 TlŽ. s 222 . . Žcf. Greenw 7,Ž. 4.3 , Example 2x. . Again this depends on the choice of T . Ž.4.2 Standard Basis Elements. We say that a basis element i, l of SQޚŽ.has weight Ž, .if wt 00 Ž.i s and wt Ž.l s . Just as in Woodcock wx17 , the semi-standard -tableaux T of weight are in bijective corre- spondence with the basis elements i, l of weight Ž., . The correspon- dence is given by Tiil ,lŽ.. This does not depend on the choice of T : if we replace T by T for some g ᑭ Q , then i and lŽ. are replaced y1 y1 by i and lŽ. , respectively, and i, lŽ . is unchanged. We define a PARTITION ALGEBRAS 109 basis element i, l to be left standard if it corresponds under the above bijection to a standard tableau. We define i, llto be right standard if ,iis left standard. We extend these notions to TA by declaring a basis element i, l of TA to be leftŽ. resp. right standard if iŽQ.Ž, j Q.is so for all Q 4 0. The proof of the following lemma is left to the reader. Ž.4.3 LEMMA. Take i, l g I Žގ0 , ގ ., and put s wt Ž.i , s wt Ž.l . Take Qgގwith , g ⌳wxQr2. Then i, l is leftŽ. resp. right standard if and only if iŽQ.Ž, j Q.is so. The definitions of standardness transfer in the obvious way to SQkŽ. and Tk . Ž.4.4 Codeterminants. Following Greenwx 8 we define a codeterminant of q shape g ⌳ 0 Ž.Q in SQޚ Ž.to be an element of the form i, ll,jwhere q wt 0Ž.l s . Similarly a codeterminant of shape g ⌳ in TA is an element of the form i, ll,jiwhere wtŽ.l s . In either case ,ll,jis a standard codeterminant if i, llis left standard and ,jis right standard. We will refer to the following result as the straightening theorem: Ž.4.5 THEOREM. The standard codeterminants form an A-basis of TA. More precisely, an arbitrary codeterminant of shape can be uniquely expressed as a linear combination of standard codeterminants of shapes for G . Proof. Fix , g ⌳; it is enough to prove the corresponding statement within TA . Let M be the matrix over A whose columns are the coeffi- cients of the standard codeterminants in TA with respect to the basis of i, j’s in TA . ByŽ. 3.9 and Ž. 4.3 , if Q 4 0, the evaluation f ¬ fQŽ.takes ŽQ. ŽQ. codeterminants in TA to codeterminants in SQޚŽ. , and faithfully reflects the standardness property. By Greenwx 8, Sect. 6 the standard codeterminants form a basis of the latter module, so the integral matrix MQŽ.is square with determinant "1 for Q 4 0. Thus M is square with determinant "1, and the standard codeterminants form a basis of TA . For the refined statement, take a codeterminant g TA of shape , and write it as a linear combination of standard codeterminants of various shapes. On specialisation at Q 4 0, the coefficients of all terms with shapes h are zero bywx 8, Sect. 6 again, so these coefficients are zero in TA . Ž.4.6 Remark. One can check that the straightening algorithm for codeterminants given for the Schur algebra inwx 17 leads without difficulty to a corresponding algorithm for Tk . 110 MARTIN AND WOODCOCK If k is an A-algebra which is a field, the algebra Tk is very close to being quasi-hereditary. The obvious way to define a suitable notion would be to require that Tk-Mod be a highest weight category in the sense of Cline, Parshall, and Scottwx 5 . However, Tk-Mod is usually not a locally finite category, so the standard definition is not applicableŽŽ. see 5.8 for the failure of local finiteness. . We will content ourselves here with proving the basic representation-theoretic facts which exhibit the analogy; we do not formulate an appropriate axiomatisation. Choose a labelling Ž.1, Ž.2,... of ⌳q such that Ž.i G Žj .implies i F j. For example, the following ‘‘degree-wise’’ lexicographic order would do: Ž.Ž.Ž.Ž0,1,2,12 .Ž.Ž,3,2,1,1 .Ž3 .,... . Let Jikbe the ideal of T generated by the idempotents Žj.for j F i. Thus J123- J - J - иии ,1Ž.8Ј and the union of the ideals Jikis the whole of T . By the straightening theorem, Ji has a basis consisting of all standard codeterminants of shape Ž. jifor j F i. Thus the quotient J rJiiy1)1 has a basis consisting of the images of all standard codeterminants of shape i. We define the standard or Weyl module ⌬ kkŽ. to be the quotient of T by the submodule generated by all weight spaces Tk for ) . Another application of the straightening theorem shows that ⌬ kŽ. has a basis consisting of the images of those standard basis elements of the form i, lŽ.. In particular ⌬ kŽ. is infinite dimensional unless s Ž.0 . We define X the right-handed version ⌬ kŽ. analogously; the corresponding basis state- ment holds. It follows that multiplication induces an isomorphism of Tk-bimodules ⌬ ⌬X Љ kiŽ.m kki Ž.( J iirJ y1.18Ž. Ž.4.7 THEOREM. For g ⌳, Tk has a finite filtration with sections of the form ⌬ kkŽ. for G . The section ⌬ Ž. occurs only once, and as the top section. Proof. Taking right -weight spaces inŽ. 18Ј gives a filtration of Tk which has sections of the form ⌬ kŽ. by Ž 18Љ .. The number of times that X ⌬kkŽ.appears is the k-rank of ⌬ Ž. . By construction this is finite, and X non-zero only if F . The final statement holds since ⌬ kŽ. has rank one. Now suppose that k is a field. The usual highest weight argument shows that ⌬ kŽ. has a unique simple quotient, which we denote Lk Ž. . This is again infinite dimensional in general. PARTITION ALGEBRAS 111 Ž.4.8 THEOREM.i Ž.Ž.Lkkis characterised among the simple T -modules by the properties of ha¨ing dim LkkŽ. s 1 and L Ž. s 0 for all g . q Ž.ii The Lk Ž. for g ⌳ form a complete set of pairwise non-isomor- phic simple modules in Tk-Mod. Proof. Ž.iIfLis any simple module with these two properties there is a surjective map Tk ª L which kills the kernel of the projection onto ⌬ kŽ. . Thus L ( LkŽ. . Ž.ii Let L be any simple module. By definition of Tk-Mod, L is the sum of its weight spaces, so taking with L / 0 gives a surjection Tkkª L. Since T ( Tkwhenever and are conjugate under the action of ᑭϱ, we may assume that is dominant, and moreover that it is maximal among the dominant weights of L. The proof ofŽ. i gives L ( LkŽ.. Ž.4.9 Remark. For V g Tk-Mod and g ⌳ we define the composition multiplicity of LkŽ. in V by V : LkkŽ. s V : L Ž. , where the multiplicities on the right are computed in the category of modules for the localisation TkŽÄ4 .. The latter category is locally finite because TkŽÄ4 .is a finite dimensional algebra, so the notion of composi- tion multiplicity is well-defined there. The composition multiplicities are finite whenever V has finite dimensional weight spaces; this holds in particular for the standard modules. Ž.4.10 THEOREM. Suppose that k is a field and that ⌫ is finite. Suppose q q q further that ⌫ [ ⌫ᑭϱ l ⌳ is a coideal in ⌳ for the dominance order. Then the isomorphism classes of simple TkŽ.⌫ -modules are naturally indexed q by ⌫ , and TkŽ.⌫ is a quasi-hereditary algebra with respect to the dominance order on ⌫q. Proof. By Morita equivalence we may replace ⌫ by ⌫q.If g⌫q we q certainly have LkkŽ. / 0. On the other hand if g ⌫ with L Ž. / 0, then F ,sog⌫q. This establishes the claim about the simple modules. Quasi-heredity follows by localisingŽ. 4.7 . The standard modules ⌫ q are the ⌬ kŽ. for g ⌫ . Ž.4.11 Remark. Any finite subset ⌫ of ⌳qis contained in a finite coideal, since G implies that << F < <. In view of the above theorem, one might thus say that Tk-Mod is ‘‘residually a highest weight category’’ when k is a field. 112 MARTIN AND WOODCOCK 5. POSITIVE CHARACTERISTIC Ž.5.1 If char k s 0, any ring homomorphism : A ª k factors through ޑwx¨, so is uniquely determined by the element Ž.¨ g k. Moreover any element of k determines a homomorphism in this way. When char k s p ) 0 things are rather different. Denote by ޚ p Žresp. ޑp.Žthe ring of p-adic integers resp. numbers. , considered as topological rings with respect to the p-adic topology. For R g ޚ p we have a unique ring map Rp: ޑwx¨ ª ޑ sending ¨ to R. We claim that RpŽ.A F ޚ . For fixed i 0 the map ޑ ޑ , g gis polynomial, hence continuous. G ppª ¬ ž/i We can write R s lim Rmmwith the R g ޚ, thus ª ¨RRm Rpsslim g ޚ , ž/ii ž/ ªž/i establishing the claim. Composing with the projection ޚ ppª ކ and the inclusion ކp ª k gives a specialisation A ª k which we also denote R. Ž.5.2 LEMMA. Let k be a field of characteristic p ) 0. Ž.i E¨ery ring map : A ª k is of the form R for a uniquely determined R g ޚ p. In particular its image is the prime subfield of k. l Ž.ii If N - p , the restriction of RNto A depends only on the first l terms of the p-adic expansion of R. Proof. The product¨ ¨ is the number of ways to choose an ordered ž/i ž/j pair of subsets I, J ofÄ4 1, . . . , ¨ such that < i ¨ ¨ i i q j y l ¨ s Ý .19Ž. ž/ilž/ž/ž/ji ž/ iqjyl ls0 Put Apps Arp A,anކ-algebra. UsingŽ. 19 and the p-adic expansion of the binomial coefficient, we have in Ap n p j n ¨¨¨¨q j jjsqj,pFn.20Ž. ž/pnž/nnž/pj žqpp / ž/ž/ r Write n s n01q npqиии qnprjwith 0 F n F p y 1. If all n jare strictly less than p y 1 we have in ކp j n q p nj q 1 ssnj q1/0. ž/pj ž/1 PARTITION ALGEBRAS 113 Induction usingŽ. 20 shows that quite generally, ¨ ¨ belongs to the subring of A generated by 1 i r if n p r. ž/n p ½5ž/pi F F F Ž.21 UsingŽ. 20 again we have in Ap ¨¨¨ ¨ jjsŽ.iq1 jqi j Ž.1Fi-p. ž/ž/pipž/Ž.iq1p ž/ ip By induction on i we deduce the following equality in Ap: ¨ ¨ m Ž.i 1! Ž.j 0, 1 i - p . Łpjy s q i1pj G F 0FmFiž/ž/ ž/Ž.q Ž.22 Let : A ª k be a ring map; then factors through Ap, so byŽ. 22 with ip1 we have ¨¨ކ for all j 0. Write R with 0 R syž/ppjjg pjG ž/s F j Fpy1 and put R s R01q Rpqиии g ޚ p. We have r R01qRpqиии qRpr sRjpin ކ whenever r G j, ž/pj thus by continuity ¨¨ R.2Ž.3 Rjž/ppjjss ž/ The stated results now follow fromŽ. 21 and Ž. 23 . Let k be a field of characteristic p ) 0. Take R g ޚ p and regard k as an A-algebra via the homomorphism R Ž.5.1 . From Ž. 3.9 we get the following useful isomorphism, which shows that in positive characteristic we may view Tk as being pasted together from ordinary Schur algebras. Ž.5.3 PROPOSITION. Let ⌫ ; ⌳ be a finite set of weights, and take Q, l g ގ satisfying g⌳wxQr2 ᭙g⌫ l i-p ᭙g⌫᭙iG1 Q'R mod pl. 114 MARTIN AND WOODCOCK Then the specialisation f ¬ fQŽ. of Ž3.9 .induces an isomorphism of k- algebras ŽQ. TkkŽ.⌫ ( S Ž.⌫ , Q . Proof. The condition on Q ensures that ⌫ ; ⌳wxQ , so the result follows fromŽ. 3.9 and Ž.Ž. 5.2 ii . Note that there are always infinitely many pairs Ž.Q, l satisfying the hypotheses of the proposition. As an immediate corollary we read off the connection between PkŽ.k and the Schur algebras which was announced in the Introduction. We also reduce the problem of the composition multiplicities of the standard modules for Tk in positive characteristic to a subproblem of the analogous problem for the general linear groups. Ž.5.4 COROLLARY. Let k be as abo¨e. Ž.i Choose Q, lasin Ž5.3 . for ⌫ s Hnwx_ÄŽ.0.4 Then the partition ŽQ. algebra PkkŽ.n is Morita equi¨alent to S Ž⌫ , Q.. Ž.ii Choose Q, lasinŽ.5.3 for ⌫ s Ä4, . Then ŽQ.ŽQ. ⌬kkŽ.:L Ž .s⌬ k Ž .:L kŽ.,2Ž.4 where the composition multiplicities on the right are as defined in Ž.4.9 , and the ⌬ and L on the right are the Weyl and simple modules for the appropriate GLnŽ. k . Let k be as above. We now describe the simple modules for Pk and PkkŽ.n. By Ž 3.11 . those for P are the non-zero images under localisation q of the simple Tk-modules, so are indexed by the weights g ⌳ with H LkŽ./0 Ž see Ž 3.8 . for notation . . Similarly, for fixed n G 1 the simple q K PkŽ.n-modules are indexed by those g ⌳wxn l ⌳ such that LkŽ. / 0, where K s Hnwx_ÄŽ.0.4 Recall that a dominant weight is said to be column p-regular if all the differences iiy q1are strictly less than p. The following theorem is an analogue of Greenw 7,Ž. 6.4bx . q H Ž.5.5 THEOREM.i Ž.For g ⌳ , we ha¨eLkŽ./0if and only if is column p-regular. q K Ž.ii For g ⌳wxn l ⌳ , we ha¨eLkŽ./0if and only if is column p-regular, and / Ž.0 if R g pޚ p. Proof. Ž.i After a few preliminary reductions the proof follows that of w7,Ž. 6.4bx , and for the most part we only point out how to make the appropriate modifications; unexplained notation is taken fromw 7,Ž. 6.4bx . PARTITION ALGEBRAS 115 Fix g ⌳q and suppose that is not column p-regular. Since the set of weights of any Pk-module is closed under the natural action of ᑭϱ,itis a enough to show that LkŽ. s 0 for every weight of the formŽ 1 . Ž.aG0 . Take Q, l g ގ satisfying the hypotheses ofŽ. 5.3 with ⌫ s Ä4, , and choose n so that and have less than n non-zero parts. By the conclusion ofŽ. 5.3 we may work instead in the ordinary Schur algebra ŽQ.ŽQ. SnkŽ.,Q. The weights and are replaced by s and s s a ŽQya, 1.Ž , and the module Lkkk .Žby the simple Sn,Q.Ž-module L .of highest weight . Let T s T be the Young diagram of , and l s lŽ. the canonical indexŽ. 4.1 . Let RT Ž.Žresp. CT Ž..be the subgroup of ᑭ Q preserving the rowsŽ. resp. columns of T. The weight space in question is spanned by the elements ÝŽ.Tlh: T Ž.25 hgiRŽ. T for i ranging over all indices of weight w7,Ž. 6.4cxw . By 7,Ž. 5.3bx we may further restrict ourselves to those indices i with Ti standard, so 0 иии 0 )) иии ) ) иии )) Tis иии ). ) . X where all the elements ) are distinct and non-zero. Let T denote the tableau obtained by deleting the first row of T. Let K be the subgroup of RTŽ.consisting of those elements which fix the first row pointwise and preserve the set of columns of T X. Then K acts fixed point freely on iRŽ. T , and its order is Ž.Ž. 1223y ! y ! иии . It follows that the sum Ž. 25 is zero by breaking it up into K-orbits. Conversely, suppose that is column p-regular. In this case we choose << Q<< as<<, so that s ŽQ y << , 1.Ž . Let u be the index 0y , 1, 2, . . . , << .. We claim that the element of LkŽ. given by u,lllи Ž.T : T s ÝÝsgnŽ. c l,u,26 Ž. X gCTŽ.gRT Ž. is non-zero, where RTXŽ.is the subgroup of RTŽ.fixing the first row X pointwise. Let g RTŽ.be the permutation which reverses the entries in X the columns of T , and consider the coefficient of cl , u inŽ. 26 . The argument ofwx 7, p. 96 shows mutatis mutandis that if g CTŽ.and X gRTŽ.with cl , uls c ,u, then s . The subgroup of elements XŽ. ŽŽ ..q gRT which fix l has order w s Ł q G1 qqy q1! , so the re- quired coefficient is "w / 0. 116 MARTIN AND WOODCOCK Ž.ii If Ž. 0 / g ⌳qwxn the statement follows from the proof ofŽ. i , a since the demonstration of non-vanishing uses the hookŽ 1. with a s << . For s Ž.0 , however, the proof of non-vanishing uses the zero hook. By Ž.2.9 we lose a simple module precisely when ¨ is zero in k, and the only one that we can lose isŽ. 0 . Ž.5.6 Remark. Let k be a ޚwx¨ -algebra which is a field of characteristic p ) 0, and let R be the image of ¨ in k.If Rdoes not lie in the prime ކ Ž. n subfield pkof k, P n is Morita equivalent to Ł is0kᑭ iwx15 , so in this case understanding the representation theory of the partition algebras is equivalent to understanding that of the symmetric groups. Note that the simple modules are again indexed by the column p-regular partitions of 0, 1, . . . , n Žsee, e.g.,w 7,Ž. 6.4bx. . On the other hand, if R g ކp we can choose R g ޚ p congruent to R modulo p, make k into an A-algebra via R , and applyŽ. 5.4 . Ž.5.7 Decomposition Matrices. For R g ޚ, PޚŽR.Ž.n is a ޚ-order in PŽR.Ž.nwhose reduction modulo p is P ŽR.Ž.n . Take R Ä0, 1, . . . , 2n ޑކpf y 14 so that Pޑ ŽR.Ž.n is semisimplewx 14Ž. see also the Appendix below . For , g ⌳qwxn , with column p-regular, and non-zero if R is congruent to zero modulo p, let d, be the composition multiplicity of the simple P ŽR.Ž.n -module indexed by in the reduction modulo p of ކ p the simple Pޑ ŽR.Ž.n -module indexed by . Thus Ž.d, is the decomposi- tion matrix for the triple ŽŽ.Ž.Ž..P ŽR.Žn , P R.Žn , P R.n . Choosing Q and l ޑޚކp as inŽ. 5.4 we have d⌬ŽQ.Ž:LQ.. ,sކކppŽ. Ž. The flow of information is in this case reversible: if one knows the decomposition matrices for all the PkŽ.n one can read off the decomposi- tion matrices for all the symmetric groups. As Erdmann has shownwx 3 , the latter information allows one in principle to recover the composition multiplicities of all Weyl modules for all general linear groups. Ž.5.8 The Rank One Case. We now give an extended example which elucidates the structure of the Weyl modules in the single-rowed localisa- tion of Tk in positive characteristic. This is closely connected to the modular representation theory of SL2Ž. k . Corollary Ž 5.10 . below shows that in general the Weyl modules for Tk have infinite length, and hence that Tk-Mod is not a locally finite category. Let k be a field of characteristic p ) 0, made into an A-algebra via the specialisation f ¬ fRŽ.for some R s R01q Rpqиии g ޚ p. Let ⌫ be the set of single-rowed weights in ⌳, i.e., those of the form Ž.n for some n G 0. PARTITION ALGEBRAS 117 Note that ⌫ is a coideal in ⌳. Fix n G 0; in order to understand the ⌫ structure of ⌬ kŽ. , we work in the localisation at the finite set of weights ÄŽ.Žn,nq1 . ,..., Žnqm .4where m G 0 is fixed but arbitrary. Choose l so ll1l that p G m q n; then p qG 2 p G 2Ž.Ž.n q m , so by 5.3 the information we require is faithfully represented in the Schur algebra SkŽ.2, Q , where l lq1 Q s R01q Rpqиии qRplqp . Under this correspondence the weight Ž.n q i g ⌳ for 0 F i F m corre- sponds to the weight Ž.Ž.Q y n y i, n q i for Sk 2, Q . Passing to the alge- braic groupŽ.Ž. scheme SL2 k and using the identification Ž.a, b l a y b, this becomes N [ Q y 2Ž.n q i . Let us recall some facts from the representation theory of SL2Ž. k . The Weyl module ⌬ kŽ.N for SL201 Ž. k has a standard basis ¨ , ¨ ,...,¨Nof weight vectors of weights N, N y 2,...,yN, respectively, on which the Ž j.Žj. standard generators e , f of the hyperalgebra of SL2Ž. k act Ž for j ) i . by j Ž jyi. f ¨ijs ¨ ž/jyi Žjyi. Nyi e ¨jis ¨. ž/jyi Let ) be the least reflexive transitive relation onÄ4 0, . . . , N defined by j ) i if either j j ) i and / 0inކp ž/jyi or N y j j - i and / 0inކp, ž/iyj and let ; be the equivalence relation onÄ4 0, . . . , N defined by i ; j if Ž j. and only if i ) j and j ) i. From the description of the action of e and Ž j. f on ⌬ kŽ.N we see that Ž.i The submodules of ⌬ kŽ.N are the subspaces of the form ⌬kiŽ.N⌶[kÄ4¨Nig⌶ for ⌶ an ideal in the posetŽÄ4 0, . . . , N , ).. Ž.ii The composition factors of ⌬ k ŽN .are in bijective correspon- dence with the equivalence classes of ; . 118 MARTIN AND WOODCOCK Letting m, and hence also l and Q, tend to infinity we obtain the following description of the Weyl modules for the algebra TkŽ.⌫ . Let ) be the least reflexive transitive relation on ގ0 defined by j ) i if and only if j j ) i and / 0inކp ž/jyi or R y 2n y j j - i and / 0inކp, ž/iyj and define ; as before. Let ui be the basis vector of weight n q i in the Weyl module ⌬ kkŽ.Žn for the localisation of T at the set of single-rowed partitions. . Ž.5.9 LEMMA.i Ž.uji belongs to the submodule generated by u if and only if j ) i Ž.ii The submodules of ⌬ k Ž.n are the subspaces of the form ⌬kiŽ.n⌫[kuÄ4Nig⌫ for ⌫ an ideal in Ž.ގ0 , ) . Ž.iii The composition factors of ⌬ kŽ.n are in bijecti¨e correspondence with the equi¨alence classes ⌶ of ; . Gi¨en such a class ⌶, the character of Žnqi. the associated simple module is Ýig ⌶ e . Write ¨ pŽ.i for the p-adic valuation of the number i. It is routine to check that in the special case where R s n s 0 we have j ) i if and only if ¨ppŽ.jF¨ Ž.i. Ž.5.10 COROLLARY. In the case R s 0 we ha¨e Ž.ichLp Ž l. Ý eŽi.. s ¨pŽi.sl Ž0. Ž.ii ch L Ž.0 s e . Ž.iii The submodule structure of ⌬ k Ž.0 is иии ⌬ 0 s Vy101; V ; V ; ; Vϱ; kŽ.0, where V V LpŽi.Ž,V V,and ⌬ 0.V LŽ0.. iir y1( ϱsDigގ0ikr ϱ( 6. CHARACTERISTIC ZERO Finally we analyse the representation theory of Tk in characteristic zero. Recall that for any A-algebra k, the localisation of Tk at the set of hook q weights H is Morita equivalent to the global partition algebra Pk Ž.3.11 . PARTITION ALGEBRAS 119 Ž.6.1 PROPOSITION. Ifkisan A-algebra which is a field of characteristic zero, Tkkis Morita equi¨alent to P . q Proof. By the preceding remarks it suffices to show that H V / 0 for all non-zero V g Tk-Mod, and since every non-zero module has a composition factor, we may assume that V s LkŽ. is simple. ByŽ. 3.11 the localisation q of Tk at Hnwxis Morita equivalent to the algebra PkŽ.F n . Since any g⌳q belongs to some ⌳wxn , it is enough to show that the number M of q simple PkŽ.F n -modules is N s Ž.6.3 THEOREM. Let k be as abo¨e and suppose that the image of ¨ is not in ގ0. Then Tkkand P are semisimple. Proof. The two algebras are Morita equivalent, so we treat Pk . Let eikgPbe the idempotent defined in the proof ofŽ. 2.9 , and put fns e0 qиии qennkn. Then f P f is the algebra PkŽ.F n , and this is semisimple by Ž.2.9 andwx 14Ž. see also the Appendix below . It suffices to show that Pkie is a semisimple module for all i G 0. If n G i we have op End f P e e P e End P e . PkŽFn.Ž.Ž.nki( iki ( Pk Ž.ki Thus fnkiP e and P kie have the same number of indecomposable sum- mands. The former is semisimple, however, and since any simple subquo- tient of Pkie is detected in fnP kie for large enough n, we infer that Pkie is itself semisimple. Ž.6.4 THEOREM. Let k be an A-algebra which is a field of characteristic q zero in which ¨ maps to R g ގ0. For g ⌳ let i g ގ be minimal such that Rqi)<<qi,2Ž.7 and define ˆ << sŽ.1,..., iy1,Rqiy , iq1,... . Ž.28 If ˆis dominant, the standard module ⌬ kkŽ. for T has length two and socle LkkŽ.ˆˆ.If is not dominant ⌬ Ž. s Lk Ž. . This has a pleasant pictorial interpretation. Assign to each point Ž.x, y gގ=ގthe value << q y y x, and let Ž.i, j be the lexicographically least 120 MARTIN AND WOODCOCK point where the value R appears outside the Young diagram of . Then ˆ is the weight with Young diagram wx j Ä4i = Ä1, 2, . . . , j 4. For example, if sŽ.4, 2, 2, 0, . . . we see from the diagram IIII12 13 иии II 9 101112 иии II 8 9 10 11 иии 567 8910иии that ⌬ kŽ. is simple if and only if R g Ä40, 1, 2, 3, 4, 7, 8, 11 . Proof. Let k and R be as in the statement ofŽ. 6.4 . We prove the theorem by applying the Jantzen sum formula in positive characteristic. For fixed n g ގ let ⌫ : ⌳ be the set of dominant weights of degree at most n. By localisation, the theorem holds for ⌬ kŽ. if and only if the ⌫ analogous statement holds for the TkkŽ.⌫ -module ⌬ Ž. for all n 4 0. Take n large enough so that and ˆŽ.whenever it is dominant belong to ⌫. If A is a ring which is free of finite rank over ޚ, and Lޑ is an absolutely irreducible Aޑޑ-module, the reduction modulo p of L isŽ. absolutely irreducible for all sufficiently large primes p. This observation is no doubt well known, but as we could not find a suitable reference, we give an argument here. Let Lޚޑbe a ޚ-lattice in L invariant under A, and let ¨1,...,¨n be a ޚ-basis of Lޚޑ. Since L has only scalar endomorphisms, the Jacobson density theorem implies that there are elements ei, j in Aޑ y1 with ei, jl¨ s ␦ j,li¨ . We can write e i,jis a ,jif ,jiwhere 0 / a ,jg ޚ and fi, j gA.If pis a prime which does not divide any of the ai, j, the reduction modulo p of Vޚ is absolutely irreducible. Ž R. Put ކ s ކpk. Since T Ž.⌫ is finite dimensional and all its simple mod- ules are absolutely irreducible, for large primes p, the simple modules for Tކ Ž.⌫are obtained by reduction modulo p of those for Tk Ž.⌫ . In proving the theorem we may thus replace k by ކ. Fix such a prime p G 2n; then the conditions ofŽ. 5.3 are met with Q s R q p and l s 1, giving an isomorphism of ކ-algebras ŽQ. TކކŽ.⌫ ( S Ž.⌫ , Q .29Ž. The right hand side is a localisation of the Schur algebra Snކ Ž.q1, Q and we can apply standard results from the modular representation theory of algebraic groups. ކ Let G be the -group scheme GLnq1. To be consistent with the notation of Section 3 we index the rows and columns of matrices and entries of weights by the numbers 0, 1, . . . , n. Let²: , denote the canonical inner nq1 product on the weight lattice ޚ of G, and 0 ,...,n the standard PARTITION ALGEBRAS 121 orthonormal basis. The Weyl group is W s SymŽÄ4 0, 1, . . . , n .. The set of positive roots is Ä4ijy N 0 F i - j F n . Set is 0y iŽ.i s 1,...,n ; the reflection perpendicular to i is the transpositionŽ. 0 i . Put s n1 Ž.n,ny1, . . . , 0 and define the ‘‘dot action’’ of W on ޚ qby w и s wŽ. q y . nq1 For a dominant weight g ޚ let ⌬ކ Ž. denote the Weyl module for G l of highest weight , and let ⌬ކ Ž. denote the lth term in its Jantzen 1 filtrationwx 9, II.8 . In particular ⌬ކކŽ. is the radical of ⌬ Ž. . ŽQ. Put s . It is easy to check that in this case the Jantzen sum formulawx 9, II.8.19 reduces to l ÝÝch ⌬ކŽ. s Ž.Ž.0 j и q pj.30Ž. l)01FjFn ²:j,q)p Here is defined by Ž.sch ⌬ކ Ž. if is dominant Ž.wиssgn Ž.Ž.w ᭙wgW. Note that if t и s for some transposition t g W we have Ž.s0. For j g n put и  << Ž.js Ž0j . qpjjsŽ.yjqp,1,..., jy1,Rqjy , jq1,... . Ž.31 We have ²:j, q s 0 y jjq j s p q R q j y << y , so there is a term inŽ. 30 corresponding to j if and only if Rqj)<<qj.3Ž.2 IfŽ. 32 fails for all j, ⌬ކ Ž. is simple. If not, let i be the least j for which Ž.32 holds. Then Ž.i fails to be dominant if and only if << Rqiy1s qiy1.3Ž.3 Suppose that this is the case and take n G j G i. Then fromŽ. 31 we have и << Ž.ij Ž.j sŽ...., iy1,Rqiy ,... sŽ..., iy1, iy1q1,.... , and the latter weight is fixed under the dot action of Ž.ŽŽ..i y 1 i ,so j s0 for all j G i. Thus the right hand side ofŽ. 30 is zero and ⌬ކ Ž. is again simple. 122 MARTIN AND WOODCOCK ŽQ. Finally suppose that ˆˆs Ž.i is dominant. Note that s ˆ , and by choice of n this is again in ⌫. We consider the sum formula for ˆ: ²:ii,ˆqsŽ.Ž.yiqpyRqiy<<qi spyŽ.Rqiy<<yi-p ²: iq1,ˆqsŽ.iiyiqpyq1qiq1spqiiyq1q1)p. Thus j gives a contribution to the sum formula for ˆ if and only if j ) i, and in this case the corresponding weight is ˆŽ.jsŽ.jiyjqp,...,Qqiy<<,..., qjyi,... ,Ž. 34 where the displayed entries are in positions 1, i, and j, and if l f Ä41, i, j the lth entry is l. ComparingŽ 31 . and Ž 34 . we see that ˆ Ž.j s Žij .и Ž.j , which gives l l l ÝÝÝch ⌬ކކŽ. s ch ⌬ Ž.ˆˆˆˆy ch ⌬ ކކ Ž. s ch L Ž. y ch ⌬ ކ Ž. . l)0 l)0 l)1 Since the left hand side is a non-negative integral combination of charac- 1 ters of simple modules we see that we must have ⌬ކކŽ. s L Ž.ˆ . Ž.6.5 Remarks. Ž. i Theorem Ž. 6.4 yields enough information to deter- mine the complete structure of Tk-Mod for k as given. For the details cf.wx 12 . Ž.ii The exceptional structure of the partition algebras follows from Ž.6.4 by localisation. This is proved by a quite different method inwx 12 . The present proof is more conceptual, but of course the hard work is hidden in the Jantzen sum formula. APPENDIX For the convenience of the reader we sketch here the main result that we have taken from the literature on the partition algebra, namely the semisimplicity of PkŽ.n for Q f ގ0 and k of characteristic zero. 0 We first recall some elementary facts. Let e g EŽ.n be the discrete X 0 0 0 equivalence relation on n j n and ⌬ nks P Ž.ne. Note that EŽ.n l ⌬ nis 0 a basis for ⌬ n. Using this basis both the degree, l ⌺ , of the polynomial 0 determinant of the Gram matrix of ⌬ n, and the maximum l00g ގ such that Ql0 divides this polynomial are readily computed. On the other hand we know several factors of this polynomial from the non-semisimplicity of 0 ⌬n for Q g Ä41, 2, . . . , n y 1ŽŽ.. see 6.4 . The total degree of these factors is PARTITION ALGEBRAS 123 0 l ⌺ y l0 , thus the polynomial is non-vanishing for Q f ގ0 , and thus ⌬ n is simple projective in these cases. X X X X Let e g EŽ.n beÄÄ 1, 14Ä , 2, 24Ä , . . . , Ž.Ž.n y 1, ny1,4Än 4Ä,n44. Then we have isomorphisms of algebras e PkkŽ.ne(P Žny1 . PkkkŽ.nrP Ž.neP Ž.n (kᑭ n.3Ž.5 Note that if ⌬ is a simple projective left PkŽ.n -module then e⌬ is either a simple projective left e PkŽ.ne-module, or zero. For a dominant weight let ⌬ nkbe the corresponding standard P Ž.n -module. It is readily verified that as a PknŽ.n y 1 -module ⌬ has a filtration with sections of the form ⌬ ny1, where the degree of differs from that of by no more than one, and that P Ž.n ⌬ has a filtration with sections of the form k mPk Žny1. ny1 ⌬ n , where the degree of differs from that of by no more than one. In 0 particular if m - n, ⌬ nkhas a filtration as a P Ž.m -module with sections of the form ⌬ m, in which every with 0 F << F n y m appears. Suppose that PkŽ.m y 1 is semisimple. Then by elementary category theory argu- ments a PkmŽ.m -module morphism ⌬ ª ⌬ mimplies & m and & q for some q - m. Now consider the adjoint isomorphism Hom ⌬ , ⌬0 Hom P n ⌬ , ⌬0 . PkkŽm.Ž.mn( PŽn.Ž.kŽ.mPkŽm.mn Note from the above that when Q f ގ0 the right hand side is empty if <<)nym. Thus if Ä⌬ rk<< <- l4 are simple projective for P Ž.r then Ä⌬<< 4 Ž. Ž. ry1 N-lq1 are so for Pkkr y 1 , since as a P r y 1 -module ⌬0 Ä⌬<< 4 ŽŽ. lqrris a direct sum with N s l q 1 among its summands. 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