TENSORIAL SQUARE OF THE HYPEROCTAHEDRAL GROUP COINVARIANT SPACE FRANC¸OIS BERGERON AND RICCARDO BIAGIOLI Abstract. The purpose of this paper is to give an explicit description of the trivial and alternating components of the irreducible representation decomposition of the bigraded module obtained as the tensor square of the coinvariant space for hyperoctahedral groups. Contents 1. Introduction. 1 2. Reflection group action on the polynomial ring 2 3. Diagonally invariant and alternating polynomials 5 4. The Hyperoctahedral group Bn 7 5. Plethystic substitution 9 6. Frobenius characteristic of Bn-modules 11 7. Frobenius characteristic of Q[x]. 13 8. Frobenius characteristic of H. 13 9. The trivial component of C 15 10. Combinatorics of e-diagrams 18 11. Compactification of e-diagrams 21 12. The bijection 22 13. A basis for the trivial component of C 23 14. The alternating component of C 24 15. Compact o-diagrams 26 References 28 1. Introduction. The group Bn ? Bn, with Bn the group of “signed” permutations, acts in the usual natural way as a “reflection group” on the polynomial ring in two sets of variables Q[x,y] = Q[x1, . . . , xn, y1, . . . , yn], (?, ?) (xi, yj) = (±x?(i),±y?(j)), where ± denotes some appropriate sign, and ? and ? are the unsigned permutations cor- responding respectively to ? and ?.
- diagonally invariant
- polynomial ring
- finite reflection
- group
- qw
- diagonal bn
- invariant subspaces
- reflection group