Chapter 1 Quantum linear groups and polynomial induction In this first chapter, we collect together all the known results about quantum linear groups and the /-Schur algebra that we will need later. Then we prove some new results about polynomial induction in quantum linear groups, generalizing results of [BKi, §2] in the classical setting. 1.1. Symmetric groups and Hecke algebras Fix an integer k 1. We write v t k if v is a composition of k, so that v — (fci, A:2,...) for non-negative integers fci, &2•• • such that k\ + & 2 + * * • = k. If in addition k\ & 2 • • •, then v is a partition of k, written v h k. The height h(v) of a composition v = (fci,^,...) N fc is the smallest integer a 1 such that fca+i = fca+2 = • • • = 0. For Ahfc, write A' for the transpose partition, i.e. the partition whose Young diagram is obtained by reflecting the Young diagram of A in the main diagonal. Let denote the usual dominance order on the set of all compositions of fc, namely, /i = (mi, rri2,...) A = (Zi, fa, . •.) if and only if Yli=i m i — Yji=i h f°r a ^ 3 — 1- We a ^so write \i A if A //, and /x A if /i A but /i 7^ A. We write E^ for the symmetric group on fc letters. For w G Sfc, £(w) is the length of u, that is, the minimal number £ of basic transpositions s i , . . . , se such that u = 5i52 .. • S£. For v — (k\,..., fca) N /c, S^ denotes the Young subgroup of E& isomorphic to S/Cl x • • • x E&a. For A, /JL N fc, the set DA (resp. D" 1 ) of elements of E& which are of minimal length in their E^/EA-coset (resp. their E^Efc-coset) gives a set of distinguished E^/E^- coset representatives (resp. EM\E^-coset representatives). We set D^\ = D~lC\D\, to obtain a set of distinguished EM\E^/E/x-coset representatives. Moreover, if both EM and E^ are subgroups of E*, for some z/ h fc, the set D^ A = Z } ^ H E„ is a set of E/i\Ezy/E \-coset representatives. We will freely use well-known properties of these distinguished double coset rep- 9

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