By Percy Alexander MacMahon

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It turns out that all these identities are consequences of the identity of algebraicity. The second kind of identities, studied by Benediktovich and Zalesskij [BZ], consists of the identities of degree m for Mn(K) which are skewsymmetric in m - 1 variables. All these identities are consequences of the standard identity S2n = O. Finally, there are also polynomial identities which are obtained from central polynomials and this will be the main topic of the next chapter. We shall give some idea how one can (in principle) perform concrete calculations with polynomial identities.

Wd(A). ),,1-12 One searches for polynomial identities in the GLd-module d).. Wd(A). As a lower bound for the multiplicities n).. one uses the S2n+2-character of P2n+2 n (s2nf obtained in [BD]. In this way, one considers 77 systems with up to 7700 unknowns and finds only sufficiently good bounds for the ranks of the systems. The number of the systems can be decreased to 33, the number of unknowns to 5775 and the computing time to two hours if one uses all the available theoretical information. We shall complete this chapter with several results on co characters of matrices.

5. (i) Let A = (Al, ... ,Ar ) be a partition ofm, T E Sm and let T = TA (T) be the corresponding Young tableau. Up to a multiplicative constant the element of K Sm eT = (sign "()(Y'( L L pER(T) iEC(T) is a minimal idempotent which generates a submodule of K Sm isomorphic to M(A). (ii) The sum of all left Sm -modules K SmeT, where T runs on the set of standard A-tableaux, is direct. \) of K Sm corresponding to A, and KSm = EB I(A). Ar-m 28 Chapter 2. ~, ... ]. -tableaux T).. (T), T E 8 m . For example, if)..

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