By James Alexander Green

The first 1/2 this booklet includes the textual content of the 1st version of LNM quantity 830, Polynomial Representations of GL_{n}. This vintage account of matrix representations, the Schur algebra, the modular representations of GL_{n}, and connections with symmetric teams, has been the root of a lot examine in illustration theory.

The moment part is an Appendix, and will be learn independently of the 1st. it really is an account of the Littelmann course version for the case gl_{n}. to that end, Littelmann's 'paths' turn into 'words', and so the Appendix works with the combinatorics on phrases. This results in the repesentation conception of the 'Littelmann algebra', that is a detailed analogue of the Schur algebra. The remedy is self- contained; particularly entire proofs are given of classical theorems of Schensted and Knuth.

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**Additional resources for Polynomial Representations of GLn**

**Example text**

If k-tableau i = (il,... , Jr) is an element iT : [k] ~ ~ T i. = x(s,t), (s,t) we often place, of by refer to of In general, i I(n,r), Ti as the entry P we denote the is not bijective. in place P , If or in the T.. 3° Bideterminants of l(n,r). formula i4 n = 3 3 4 51 / The entries We define Let and K k = (3,2,0). l for any belong be an infinite an element (T i : Tj)= We might take for our i 6 I(3,5), to the set: _3 = {1,2,3}. 3a) (T i : Tj) = Here s(o) ci,jo = c ~ s(o)ci,jo = o~C(T) is the sign of -i io " o.

Then (Tg : Ti) is equal to Let 53 ,Example 2. 3h) be the element of Tg i 1 2 2 3 3 whose l(n,r) •. ,2 , . k-tableau is I1 • J In other words, gx(s,t) = s, for all (s,t) ~ [hi. Then (Tg : T~) = C(~l)C(~ 2) ... 4. 4a') Ch, j o ~ = Z ~(Ch,i)ci, j, iEI ~(n,r), ~(n,r) As left module acting SK(n,r). ~(n,r) and belongs to the category and as right module it belongs to the analogously defined category of all right, finite-dimensional coefficient space lies in ~(n,r), F = FK we have (see ~ o Oh, j = ~ ~ SK(n,r ).

7a) follows fact that the coefficient of number of h-tableaux Ti X~ in from the first, and from the S=%(XI,... e. is precisely the weight) ~ -- this argument from the definition of (see [M, p. 42]). 7b) ~ char K = O, If is the character of an irreducible module in we deduce char K = 0, then DX, K is irreducible. ~(n,r) ~A 60 We show next that the family infinite field K, we may write is defined over {~,K } (~ Z. 3. 7c) The Z-span (i E l(n,r)) is a D~, Z to ~,Z of the elements Z-form of ~,Q. 6K : ~ , Z K~ and the maps (~ : Ti) Q The family ~,K {~,K } is Z-defined by which take each (T~ : Ti) Q ® 1K D~,K~+ k~(n,r) is also defined (Te : Ti) K.