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Tensor products and dimensions of simple modules for symmetric groups. (English) Zbl 0849.20008
Let \(K\) be an algebraically closed field of characteristic \(p>0\) and let \(D^\lambda\) be the simple module of the symmetric group \(S_n\) over \(K\), where \(\lambda\) is a \(p\)-regular partition of \(r\). Then, the dimension of \(D^\lambda\) for \(\lambda\) with at most \(n\) parts are the same as the multiplicities of direct summands of \(E^{\otimes r}\) where \(E\) is the natural \(n\)-dimensional module for \(\text{GL}_n(K)\). The problem of finding the dimension of \(D^\lambda\) is a major open problem; thus that the author has been able to calculate recursive formulae for these multiplicities in the case \(n=2\) leading to a determination of their generating functions is of some interest. She shows that these are rational functions and remarkably, explicit expressions in terms of Chebyshev polynomials are obtained. These in turn lead to explicit formulae for the dimensions of the \(D^\lambda\).

MSC:
20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory
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References:
[1] E. Cline, B. Parshall, L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine angew. Mathematik391 (1988), 85–99 · Zbl 0657.18005
[2] S. Donkin, On tilting modules for algebraic groups, Math. Z.212 (1993), 39–60 · Zbl 0798.20035
[3] K. Erdmann, Symmetric groups and quasi-hereditary algebras, in Proc. Conf. Finite dimensional Algebras and Related Topics, ed. V. Dlab, L.L. Scott (Kluwer, 1994), 123–161 · Zbl 0831.20013
[4] J.A. Green, Polynomial representations ofGL n , Lecture Notes in Mathematics830, Springer 1980
[5] J.C. Jantzen, Representations of Algebraic Groups, Academic Press 1987 · Zbl 0654.20039
[6] J.C. Jantzen,Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne, J. Reine Angew. Math.317 (1980), 157–199 · Zbl 0451.20040
[7] G.D. James,Representations of the symmetric groups over the field of characteristic 2 J. Algebra38 (1976), 280–308 · Zbl 0328.20013
[8] G.D. James,On the decomposition matrices of the symmetric groups, J. Algebra43 (1976), 42–44 · Zbl 0347.20005
[9] G.D. James, A. KerberThe representation theory of the symmetric group, Encyclopedia of Mathematics16, Addison and Wesley, 1981
[10] O. Mathieu,On the dimensions of some modular irreducible representations of the symmetric group, preprint 1994
[11] C.M. Ringel,The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z.208 (1991), 209–225 · Zbl 0725.16011
[12] T.J. Rivlin,Chebyshev polynomials: From Approximation Theory to Algebra and Number Theory, (2nd ed.) Wiley 1990 · Zbl 0734.41029
[13] M.A. Snyder,Chebyshev methods in numerical approximation, Prentice Hall 1966 · Zbl 0173.44102
[14] S. Xanthopolous,On a question of Verma, about indecomposable representations of algebraic groups and their Lie algebras, PhD thesis, London 1992
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