Stanley–Reisner rings and the occurrence of the Steinberg representation in the hit problem

Hai, Nguyen Dang Ho

Comptes Rendus. Mathématique, Tome 360 (2022) no. G9, pp. 1009-1026.

Un résultat de G. Walker et R. Wood dit que l’espace des indécomposables en degré $2^n-1-n$ de l’algèbre polynômiale ${𝔽}_2[x_1,...,x_n]$, considérée comme module sur l’algèbre de Steenrod modulo 2, est isomorphe à la représentation de Steinberg de ${\mathrm{GL}}_n\left({𝔽}_2\right)$. Dans ce travail, on cherche à généraliser ce résultat à tous les corps finis. Pour ce faire, on étudie une famille d’anneaux quotients finis ${\mathbf{R}}_{n,k}$, $k\in {\mathbb{N}}^{*}$, de ${𝔽}_q[x_1,...,x_n]$, où chaque ${\mathbf{R}}_{n,k}$ est défini comme quotient de l’anneau de Stanley–Reisner d’un complexe de matroïde. On montre aussi en utilisant un variant de ${\mathbf{R}}_{n,k}$ que la dimension de l’espace des indécomposables de ${𝔽}_q[x_1,...,x_n]$ en degré $q^{n-1}-n$ est égale à celle d’une représentation cuspidale complexe de ${\mathrm{GL}}_n\left({𝔽}_q\right)$, à savoir $(q-1)(q^2-1)\cdots (q^{n-1}-1)$.

Sur le corps ${𝔽}_2$, on établit une décomposition du facteur de Steinberg de ${\mathbf{R}}_{n,2}$ en somme directe de suspensions de modules de Brown–Gitler. Ceci suggère une décomposition du facteur stable de Steinberg de la réalisation topologique de ${\mathbf{R}}_{n,2}$ en bouquet de suspensions de spectres de Brown–Gitler.

A result of G. Walker and R. Wood states that the space of indecomposable elements in degree $2^n-1-n$ of the polynomial algebra ${𝔽}_2[x_1,...,x_n]$, considered as a module over the mod 2 Steenrod algebra, is isomorphic to the Steinberg representation of ${\mathrm{GL}}_n\left({𝔽}_2\right)$. We generalize this result to all finite fields by studying a family of finite quotient rings ${\mathbf{R}}_{n,k}$, $k\in {\mathbb{N}}^{*}$, of ${𝔽}_q[x_1,...,x_n]$, where each ${\mathbf{R}}_{n,k}$ is defined as a quotient of the Stanley–Reisner ring of a matroid complex. By considering a variant of ${\mathbf{R}}_{n,k}$, we also show that the space of indecomposable elements of ${𝔽}_q[x_1,...,x_n]$ in degree $q^{n-1}-n$ has dimension equal to that of a complex cuspidal representation of ${\mathrm{GL}}_n\left({𝔽}_q\right)$, that is $(q-1)(q^2-1)\cdots (q^{n-1}-1)$.

Over the field ${𝔽}_2$, we also establish a decomposition of the Steinberg summand of ${\mathbf{R}}_{n,2}$ into a direct sum of suspensions of Brown–Gitler modules. The module ${\mathbf{R}}_{n,2}$ can be realized as the mod $2$ cohomlogy of a topological space and the result suggests that the Steinberg summand of this space admits a stable decomposition into a wedge of suspensions of Brown–Gitler spectra.

Classification : 55S10, 55P42, 05E45

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