## 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