## DIMACS TR: 93-08

## Hilbert Series of Group Representations and Grobner Bases for Generic Modules

### Author: Shmuel Onn

**
ABSTRACT
**

Each matrix representation $\pi:G\longrightarrow GL_n(K)$ of a finite group
$G$ over a field $K$ induces an action of $G$ on the module $A^n$ over the
polynomial algebra $A=K[x_1,\cdots,x_n]$. The graded $A$-submodule $M(\pi)$
of $A^n$ generated by the orbit of $(x_1,\cdots,x_n)$ is studied.
A decomposition of $M(\pi)$ into generic modules is given. Relations between
the numerical invariants of $\pi$ and those of $M(\pi)$, the later being
efficiently computable by Gr\"{o}bner bases methods, are examined. It is
shown that if $\pi$ is multiplicity-free, then the dimensions of the
irreducible constituents of $\pi$ can be read off from the Hilbert series
of $M(\pi)$. It is proved that determinantal relations form Gr\"{o}bner bases
for the syzygies on generic matrices with respect to any lexicographic order.
Gr\"{o}bner bases for generic modules are also constructed, and their Hilbert
series are derived. Consequently, the Hilbert series of $M(\pi)$ is obtained
for an arbitrary representation.

Paper available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1993/93-08.ps

DIMACS Home Page