12.36 (1992)

Let $p$ be a prime, $V$ an $n$-dimensional vector space over the field of $p$ elements, and let $G$ be a subgroup of $\text{GL}(V)$. Let $S = S[V^*]$ be the symmetric algebra on $V^*$, the dual of $V$. Let $T = S^G$ be the ring of invariants and let $b_m$ be the dimension of the homogeneous component of degree $m$. Then the Poincaré series $\sum_{m \geqslant 0} b_m t^m$ is a rational function with a Laurent power series expansion $\sum_{i \geqslant -n} a_i (1 - t)^i$ about $t = 1$ where $a_{-n} = 1/|G|$.

Conjecture: $a_{-n+1} = r / (2|G|)$ where $r = \sum_W ((p-1)\alpha_W + s_W - 1)$, the sum is taken over all maximal subspaces $W$ of $V$, and $\alpha_W$, $s_W$ are defined by $|G_W| = p^{\alpha_W} \cdot s_W$ where $p \nmid s_W$ and $G_W$ denotes the pointwise stabilizer of $W$.