arXiv subjects

Ben Blum-Smith

Publications and source records attributed to Ben Blum-Smith.

20 records · Page 2Linked to original sources

arXiv2018

When are permutation invariants Cohen-Macaulay over all fields?

We prove that the polynomial invariants of a permutation group are Cohen-Macaulay for any choice of coefficient field if and only if the group is generated by transpositions, double transpositions, and 3-cycles. This unites and generalizes several previously known results. The "if" direction of the argument uses Stanley-Reisner theory and a recent result of Christian Lange in orbifold theory. The "only-if" direction uses a local-global result based on a theorem of Raynaud to reduce the problem to an analysis of inertia groups, and a combinatorial argument to identify inertia groups that obstruct Cohen-Macaulayness.

math.AC↗

arXiv2018

Two inquiries about finite groups and well-behaved quotients

This thesis addresses questions in representation and invariant theory of finite groups. The first concerns singularities of quotient spaces under actions of finite groups. We introduce a class of finite groups such that the quotients have at worst abelian quotient singularities. We prove that supersolvable groups belong to this class and show that nonabelian finite simple groups do not belong to it. The second question concerns the Cohen-Macaulayness of the invariant ring $\mathbb{Z}[x_1,\dots,x_n]^G$, where $G$ is a permutation group. We prove that this ring is Cohen-Macaulay if $G$ is generated by transpositions, double transpositions, and 3-cycles, and conjecture that the converse is true as well.

math.AC↗