By S. Donkin

ISBN-10: 0511600704

ISBN-13: 9780511600708

ISBN-10: 0521645581

ISBN-13: 9780521645584

This e-book specializes in the illustration conception of q-Schur algebras and connections with the illustration thought of Hecke algebras and quantum normal linear teams. the purpose is to provide, from a unified standpoint, quantum analogs of definite effects identified already within the classical case. The technique is basically homological, in line with Kempf's vanishing theorem for quantum teams and the quasi-hereditary constitution of the q-Schur algebras. starting with an introductory bankruptcy facing the connection among the normal normal linear teams and their quantum analogies, the textual content is going directly to talk about the Schur Functor and the 0-Schur algebra. the following bankruptcy considers Steinberg's tensor product and infinitesimal conception. Later sections of the ebook talk about tilting modules, the Ringel twin of the q-Schur algebra, Specht modules for Hecke algebras, and the worldwide size of the q-Schur algebras. An appendix offers a self-contained account of the idea of quasi-hereditary algebras and their linked tilting modules. This quantity could be basically of curiosity to researchers in algebra and similar subject matters in natural arithmetic.

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For a k-space X we write IX ® E for the k-space X ® E, regarded as a comodule via the structure map id 0 rE : X 0 E -> X ® E ® A. Similarly we define a right comodule E 0 IX I. We define left comodules V 0 IX I and IX 0 V in the same way. Note that the structure maps E -+ JEl ® A and V -> A 0 IV I are morphisms of comodules. We shall identify a bilinear map b : X x Y -+ k with its k-linear extension X ® Y -} k. We say that a bilinear map b : V x E -> k is a pairing if it is non-singular and (id ® b) o (rv ® id) _ (b 0 id) o (id ®rE).

Similar remarks apply to '« and so we get: (3) " = '", for a E A(n, r). Let (i, j), (s, t) E UCCandCCa, b E I(n, r). We have t lbij * S,t)(Cab) = E Sij(Cah)bst(Chb)- hEI(n,r) 2. The Schur Functor and a Character Formula 38 From this and earlier remarks one sees: (i) 4if * st = 0 unless j - s. (ii) 1 = EaEA(n,r) 4- is an orthogonal decomposition. (4) E A(n, r). Let B denote the span of the set B of elements where j has content Q, and let B' denote the span of the set B' of elements Si,j,3, where i has content a.

Then J is an M-submodule of T(E). We define A(E) = T(E)/J and write A for the multiplication in A(E). Since J is homogeneous we have an induced grading and M-module decomposition A(E) = ®r o E. The rth exterior power ArE has k-basis consisting of the elements ei, A .. A ei,, with n > i1 > > it > 1. 7]. Similarly we define K to be the ideal of T(V) generated by the elements vh, with 1 < h < n, and vivj + vjvi, with 1 < i < j < n. Then K is an M-submodule of T(V). We define A(V) = T(V)/K and write A for the multiplication in A(V).

### The q-Schur algebra by S. Donkin

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