Date: Thursday, November 26, 2020.
Time: 3:30 PM.
Zoom Meeting Link:
https://us02web.zoom.us/j/85923312360?pwd=aFk5R2Z1alJ6dEV0UlBHQkV2aU84dz09
Meeting ID: 859 2331 2360
Passcode: 392843
On Hilbert ideals for a class of $p$-groups in characteristic $p$
Mandira Mondal
Chennai Mathematical Institute.
26-11-20

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Abstract

Let $\Bbbk$ be a field, $G$ a finite group and $V$ a finite dimensional linear representation of $G$ over $\Bbbk$. The action of $G$ on $V$ induces an action on $S$, the symmetric algebra of $V^*$. We shall give a brief exposition of the classification problem of representations of finite groups for which the invariant ring $S^G$ is a polynomial ring. It remains an open problem in modular invariant theory.

Suppose $\Bbbk$ is a field of characteristic $p>0$. In this talk we shall prove for a class of $p$-groups, which we call generalised Nakajima group, the Hilbert ideal (the $S$-ideal generated by positive degree invariants) is a complete intersection; hence $S^G$ is a polynomial ring if and only if it is a direct summand of $S$ as an $S^G$-module. This settles a conjecture by Shank and Wehlau for these groups. We shall also show that the Hilbert ideal can be generated by $\dim_{\Bbbk}(V)$ homogeneous elements of degree $\leq |G|$. This bound was conjectured by Derksen and Kemper. This is a joint work with Prof. Manoj Kummini.