1.2 TH E ^-SCHUR ALGEBRA 11 (1.1c) xf = yk and yf = xk. Let v = (fci,..., ka) N A:. The above definitions generalize easily to the Young subalgebra FLV Hp^iJ^u) of Hk, which is the subalgebra spanned by {Tw \ w G S^}. Identifying Hv and Hkl ® ® if^a in the natural way, we let xv (resp. y^) denote the element xkl&- -®^ifca (resp. ykl®- -®yka) oiHv. The ff^-modules2#^ Rvxv and £/^ = i f ^ are then the trivial and sign representations of FLV. As in [DJi, §3], we define the permutation module Mv and signed permutation module Nv of Hk to be left ideals Hkxv and Hkyv respectively. By (1.1c), we have that {M")# = Nu. Moreover, Hk is a free right ^-module with basis {Tw\w e Du}. Therefore, Mv can also be defined as the induced module ind ^ THU = Hk ®HU ^HV- Similarly, Nv * indg* SHv = Hk ®Hu £Hu. Define ux to be the unique element of Dy,x such that E,y HWAEA = {1}. Note that ux is precisely the element denoted wy in [DJ4, p.258]. We need the following fact proved in [DJi, Lemma 4.1]: (l.ld) For A h k, the space yyHkxx is an F-free F-module of rank one, generated by the element yyTUxx\. The Specht module Sp is the left ideal Sp = HkyxTUxxx Observe that Spx is both a submodule of Mx and a quotient of Nx . As is well-known (see e.g. [DJi, Theorem 4.15]) if F is a field of characteristic 0 and q is a positive integer, Hk is a semisimple algebra and the Specht modules {Spx | A h k} give a complete set of non-isomorphic irreducible ^-modules. In this case, we also have the following well-known characterization of Specht modules: (l.le) For F a field of characteristic 0 and q a positive integer, Spx is the unique irreducible Hk-module that is a constituent of both Nx and Mx (having multiplicity one in each). Of course, we can take the very special case with F Q and q = 1. Then, iirQ 5 i(E/c) is just the group algebra QE^ of the symmetric group over Q and we see that the Specht modules {Spx | A h k} give a complete set of non-isomorphic irreducible QEfc-modules. We will write X(Tik) for the character ring of E& over Q. For A h fc, let feei(s fc ) (1.1.2) denote the irreducible character of the symmetric group corresponding to the Specht module Spx over Q. Given in addition fi \~ k, we write 4\(ii) f°r the value of the character jx on any element of E& of cycle-type /i. 1.2. The qr-Schur algebra Continue initially with F denoting an arbitrary ring and q G F being arbitrary. Fix h 1. We write A(/i, k) for the set of all compositions v = (fci,..., fc/J 1 k of
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