By V. G. Kac, M. Wakimoto (auth.), G. M. Piacentini Cattaneo, E. Strickland (eds.)

The major objective of those lectures is first to in short survey the elemental con nection among the illustration concept of the symmetric staff Sn and the idea of symmetric services and moment to teach how combinatorial tools that come up certainly within the thought of symmetric capabilities result in effective algorithms to precise numerous prod ucts of representations of Sn by way of sums of irreducible representations. that's, there's a simple isometry which maps the heart of the crowd algebra of Sn, Z(Sn), to the gap of homogeneous symmetric features of measure n, An. This simple isometry is called the Frobenius map, F. The Frobenius map permits us to minimize calculations concerning characters of the symmetric team to calculations regarding Schur capabilities. Now there's a very wealthy and lovely thought of the combinatorics of symmetric features that has been built in recent times. The combinatorics of symmetric capabilities, then results in a couple of very effective algorithms for increasing a number of items of Schur services right into a sum of Schur features. Such expansions of goods of Schur capabilities correspond through the Frobenius map to decomposing a variety of items of irreducible representations of Sn into their irreducible parts. moreover, the Schur services also are the characters of the irreducible polynomial representations of the overall linear staff over the complicated numbers GLn(C).

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**Topics in Computational Algebra**

The most objective of those lectures is first to in brief survey the elemental con nection among the illustration thought of the symmetric workforce Sn and the speculation of symmetric capabilities and moment to teach how combinatorial equipment that come up certainly within the conception of symmetric features bring about effective algorithms to precise a variety of prod ucts of representations of Sn by way of sums of irreducible representations.

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Ber. d. Dt. Math. Verein. 90 (1988), 155-183 [NPP 84] J. Neubiiser, H. Pahlings, W. Plesken, CAS; design and use of a system for the handling of characters of finite groups, in: Computational Group Theory (ed. M. D. Atkinson) pp 195-247, London: Academic Press 1984 [Pah 90] H. Pahlings, Realizing finite groups as Galois groups. To appear [PaP 87] H. Pahlings, W. Plesken, Group actions on Cartesian powers with applications to representation theory. 56 H. PAHLINGS J. reine angew. , 380 (1987), 178-195 [Poh 87] M.

Also the rows of the Clifford matrices are then NG(2:iN)-orbit sums of characters of N/Nzil where N z ; = [2:i,NJ. In particular the first column is known: i C(m,e)j = bi(1,1) (bi(m,e) )-1 • Example: Let us consider a split extension G = 24 : As of N = Z~ by H = As with non-trivial action of H on N. g. ) Since As has no non-trivial representation of degree less than 4 over GF(2) the action of H on N is irreducible and it follows immediately that H has one orbit (of length 15) on N\{1} or H has two orbits (of lengths 5 and 10) on N\{1}.

4 .. Let p and p' be relatively prime positive integers and let A E be regular weights. Then J-l := p' A - pA' + pAo is a regular weight. PROOF: We have to show that (J-lla:) ,E ~+, n E Z+. Then: f= ° for any a: E ~+. Let a: (J-lla:) = p'(AI,) - p(A'I,) + pn, hence p divides (AI,) if (J-lla:) 0 ,E K+. f= O,which is impossible since P~, A' E P~' = , + nK, ° < (AI,) < where p for any Let p, p' be relatively prime integers ~ h v. We define a map of p~-h v x P~' -h v into itself, denoted by (J-l,J-l') 1----+ (P"P,'), and cll,Il', <,Il' = ±1 as follows.