Algèbre, Géométrie et Topologie
Stanley–Reisner rings and the occurrence of the Steinberg representation in the hit problem
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 GL n (𝔽 2 ). 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 R n,k , k * , de 𝔽 q [x 1 ,...,x n ], où chaque 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 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 GL n (𝔽 q ), à savoir (q-1)(q 2 -1)(q n-1 -1).

Sur le corps 𝔽 2 , on établit une décomposition du facteur de Steinberg de 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 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 GL n (𝔽 2 ). We generalize this result to all finite fields by studying a family of finite quotient rings R n,k , k * , of 𝔽 q [x 1 ,...,x n ], where each R n,k is defined as a quotient of the Stanley–Reisner ring of a matroid complex. By considering a variant of 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 GL n (𝔽 q ), that is (q-1)(q 2 -1)(q n-1 -1).

Over the field 𝔽 2 , we also establish a decomposition of the Steinberg summand of R n,2 into a direct sum of suspensions of Brown–Gitler modules. The module 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.

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DOI : 10.5802/crmath.359
Classification : 55S10, 55P42, 05E45
Hai, Nguyen Dang Ho 1

1 Department of Mathematics, College of Sciences, University of Hue, Vietnam
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Hai, Nguyen Dang Ho. Stanley–Reisner rings and the occurrence of the Steinberg representation in the hit problem. Comptes Rendus. Mathématique, Tome 360 (2022) no. G9, pp. 1009-1026. doi : 10.5802/crmath.359. http://www.numdam.org/articles/10.5802/crmath.359/

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