Permutations statistics, invariants and coinvariant algebra for the Weyl group of type D.

    Riccardo Biagioli, (UQAM)

Abstract: There are several connections between permutation statistics on the symmetric group and the representation theory of the symmetric group S_n. After giving a brief survey of the known results for S_n, I will show how to generalize them to the even-signed permutation group D_n. In particular I will define a major index and a descent number on D_n that allow me to give an explicit formula for the Hilbert series for the invariants of the diagonal action of D_n on the polynomial ring. Moreover I will give a monomial basis for the coinvariant algebra R(D). This new basis leads to the definition of a new family of D_n modules that decompose R(D). An explicit decomposition of these representations into irreducible components is obtained by extending the major index on particular standard Young bitableaux.

This is a joint work with Fabrizio Caselli.

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