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The Partition Algebras and a New Deformation of the Schur Algebras

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The Partition Algebras and a New Deformation of the Schur Algebras 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. Woodcock² 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. Z The partition algebras PZw¨ 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 Zwx¨ -algebra which is a P Ž.P Ž. field, and let Pk n be the algebra obtained from PZw¨ 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 kS 0 = ??? = kS 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]. ² E-mail address: [email protected]. 91 0021-8693r98 $25.00 Copyright Q 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 N Ž.resp. N0 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 lg : n j m ª g. X Thus lg 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 l Ž.i and lg 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 lgh agrees with lg on l and with lh 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.
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