Modular matrix invariants under some transpose actions
Yin Chen, Shan Ren
TL;DR
The paper addresses the problem of computing modular matrix invariants for the action of transpose by subgroups of GL_2(F_q) on M_2(F_q). It uses explicit polynomial invariants and Noether normalization to show that the invariant rings under U_2(F_q) and SL_2(F_q) are hypersurfaces, employing a-invariant theory to determine Hilbert series without solving the full defining relations. The main results establish generating sets: for U_2(F_q) the ring F_q[M_2(F_q)]^{U_2(F_q)} is generated by f1,f2,f3,f4,h0 with a single relation f4^2 = P(h0,f1,f2,f3), and for SL_2(F_q) the ring is generated by f2,f3,g0,g1,g2 (with small-q refinements) and is also a hypersurface; the Hilbert series and Cohen–Macaulay properties are derived via a-invariant methods and Noether normalization. Together, these results provide explicit descriptions of modular matrix invariants for n=2 over arbitrary finite fields and illustrate the use of a-invariant theory in determining Hilbert series and hypersurface structure in modular invariant rings.
Abstract
Consider the special linear group of degree 2 over an arbitrary finite field, acting on the full space of $2 \times 2$-matrices by transpose. We explicitly construct a generating set for the corresponding modular matrix invariant ring, demonstrating that this ring is a hypersurface. Using a recent result on $a$-invariants of Cohen-Macaulay algebras, we determine the Hilbert series of this invariant ring, and our method avoids seeking the generating relation. Additionally, we prove that the modular matrix invariant ring of the group of upper triangular $2 \times 2$-matrices is also a hypersurface.