Permutation group invariants of exterior and superpolynomial algebras

Abstract

This thesis studies the invariant theory of exterior algebras and more generallysuperpolynomial algebras when acted upon by a permutation group. The first focus, contained in Chapter 2 is on the structure, in particular the Hilbert series, of the superpolynomial invariants and antiinvariants of wreath products P [G] = P ⋉ Gn where P ≤ Sn is a permutation group and G is a finite subgroup of GL(V ). See Theorems 1.4 and 1.5. When n varies, it turns out that there is an additional shuffle product structure inducing the structure of Sym±(Sym±(V )G) on the direct sum of the Sn[G]-invariants of Sym±(V ). See Theorem 1.6. The second focus, contained in Chapter 3, explores a phenomenon (Equation (1.20)) in which the Hilbert series of the exterior algebra (∧V )G agrees with the Poincaré polynomial of the orbit poset Bn/G obtained as a quotient of the boolean lattice Bn. In Theorem 1.11, we exhibit three families of groups for which the phenomenon holds. A proof of one of the families is deferred to an appendix. A second appendix lists groups satisfying Equation (1.20) but which do not fit into any of the families of Theorem 1.11.