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

Nguyen Dang Ho Hai

doi:10.5802/crmath.359

Algèbre, Géométrie et Topologie

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

Nguyen Dang Ho Hai ¹

¹ Department of Mathematics, College of Sciences, University of Hue, Vietnam

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1009-1026.

Résumé
Abstract

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 $R_{n, k}$ , $k \in ℕ^{*}$ , 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 $(q - 1) (q^{2} - 1) \dots (q^{n - 1} - 1)$ .

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 $R_{n, k}$ , $k \in ℕ^{*}$ , 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 $(q - 1) (q^{2} - 1) \dots (q^{n - 1} - 1)$ .

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.

Reçu le : 2021-12-21
Révisé le : 2022-03-27
Accepté le : 2022-03-29
Publié le : 2022-09-29

DOI : 10.5802/crmath.359

Classification : 55S10, 55P42, 05E45

Affiliations des auteurs :

Nguyen Dang Ho Hai ¹

¹ Department of Mathematics, College of Sciences, University of Hue, Vietnam

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2022__360_G9_1009_0,
     author = {Nguyen Dang Ho Hai},
     title = {Stanley{\textendash}Reisner rings and the occurrence of the {Steinberg} representation in the hit problem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1009--1026},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.359},
     language = {en},
}

TY  - JOUR
AU  - Nguyen Dang Ho Hai
TI  - Stanley–Reisner rings and the occurrence of the Steinberg representation in the hit problem
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 1009
EP  - 1026
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.359
LA  - en
ID  - CRMATH_2022__360_G9_1009_0
ER  -

%0 Journal Article
%A Nguyen Dang Ho Hai
%T Stanley–Reisner rings and the occurrence of the Steinberg representation in the hit problem
%J Comptes Rendus. Mathématique
%D 2022
%P 1009-1026
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.359
%G en
%F CRMATH_2022__360_G9_1009_0

Nguyen Dang Ho Hai. Stanley–Reisner rings and the occurrence of the Steinberg representation in the hit problem. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1009-1026. doi : 10.5802/crmath.359. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.359/

Bibliographie
Cité par

[1] Anders Björner Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, Adv. Math., Volume 52 (1984) no. 3, pp. 173-212 | DOI | MR | Zbl

[2] Anders Björner The homology and shellability of matroids and geometric lattices, Matroid applications (Encyclopedia of Mathematics and Its Applications), Volume 40, Cambridge Univ. Press, 1992, pp. 226-283 | DOI | MR | Zbl

[3] Edgar H. Jr. Brown; Samuel Gitler A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra, Topology, Volume 12 (1973), pp. 283-295 | DOI | MR | Zbl

[4] Michael W. Davis; Tadeusz Januszkiewicz Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J., Volume 62 (1991) no. 2, pp. 417-451 | MR | Zbl

[5] Leonard E. Dickson A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc., Volume 12 (1911) no. 1, pp. 75-98 | DOI | MR | Zbl

[6] Masateru Inoue $𝒜$ -generators of the cohomology of the Steinberg summand $M (n)$ , Recent progress in homotopy theory (Baltimore, MD, 2000) (Contemporary Mathematics), Volume 293, American Mathematical Society, 2000, pp. 125-139 | DOI | MR | Zbl

[7] Nicholas J. Kuhn; Stephen A. Mitchell The multiplicity of the Steinberg representation of ${GL}_{n} F_{q}$ in the symmetric algebra, Proc. Amer. Math. Soc., Volume 96 (1986) no. 1, pp. 1-6 | MR | Zbl

[8] Jean Lannes; Saïd Zarati Sur les foncteurs dérivés de la déstabilisation, Math. Z., Volume 194 (1987) no. 1, pp. 25-59 | DOI | Zbl

[9] George Lusztig The discrete series of $G L_{n}$ over a finite field, Annals of Mathematics Studies, 84, Princeton University Press; University of Tokyo Press, 1974 | Zbl

[10] Dagmar M. Meyer; Larry Smith Poincaré duality algebras, Macaulay’s dual systems, and Steenrod operations, Cambridge Tracts in Mathematics, 167, Cambridge University Press, 2005 | DOI | Zbl

[11] Pham Anh Minh; Grant Walker Linking first occurrence polynomials over $𝔽_{p}$ by Steenrod operations, Algebr. Geom. Topol., Volume 2 (2002), pp. 563-590 | DOI | MR | Zbl

[12] Stephen A. Mitchell Finite complexes with $A (n)$ -free cohomology, Topology, Volume 24 (1985) no. 2, pp. 227-246 | DOI | MR | Zbl

[13] Stephen A. Mitchell; Stewart B. Priddy Stable splittings derived from the Steinberg module, Topology, Volume 22 (1983) no. 3, pp. 285-298 | DOI | MR | Zbl

[14] Huynh Mui Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 22 (1975) no. 3, pp. 319-369 | MR | Zbl

[15] James Oxley Matroid theory, Oxford Graduate Texts in Mathematics, 21, Oxford University Press, 2011 | DOI | Zbl

[16] Daniel Quillen Homotopy properties of the poset of nontrivial $p$ -subgroups of a group, Adv. Math., Volume 28 (1978) no. 2, pp. 101-128 | DOI | MR | Zbl

[17] Gerald Allen Reisner Cohen–Macaulay quotients of polynomial rings, Adv. Math., Volume 21 (1976) no. 1, pp. 30-49 | DOI | MR | Zbl

[18] Larry Smith An algebraic introduction to the Steenrod algebra, Proceedings of the School and Conference in Algebraic Topology (John Hubbuck, ed.) (Geometry and Topology Monographs), Volume 11, Geometry & Topology Publications, 2007, pp. 327-348 | MR | Zbl

[19] Louis Solomon The Steinberg character of a finite group with $B N$ -pair, Theory of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968), Benjamin, New York, 1969, pp. 213-221 | Zbl

[20] Richard P. Stanley Combinatorics and commutative algebra, Progress in Mathematics, 41, Birkhäuser, 1996 | Zbl

[21] Robert Steinberg Prime power representations of finite linear groups, Canad. J. Math., Volume 8 (1956), pp. 580-591 | DOI | MR | Zbl

[22] Michelle L. Wachs Poset topology: tools and applications, Geometric combinatorics (IAS/Park City Mathematics Series), Volume 13, American Mathematical Society; Institute for Advanced Study, 2007, pp. 497-615 | DOI | MR | Zbl

[23] Grant Walker; Reginald M. W. Wood Young tableaux and the Steenrod algebra, Proceedings of the School and Conference in Algebraic Topology (Geometry and Topology Monographs), Volume 11 (2007), pp. 379-397 | MR | Zbl

[24] Grant Walker; Reginald M. W. Wood Polynomials and the $\mod 2$ Steenrod algebra. Vol. 2. Representations of $GL (n, 𝔽_{2})$ , London Mathematical Society Lecture Note Series, 442, Cambridge University Press, 2018 | Zbl

[25] Grant Walker; Reginald M. W. Wood Polynomials and the $\mod 2$ Steenrod algebra. Vol. 1. The Peterson hit problem, London Mathematical Society Lecture Note Series, 441, Cambridge University Press, 2018 | Zbl

[26] Reginald M. W. Wood Steenrod squares of polynomials and the Peterson conjecture, Math. Proc. Cambridge Philos. Soc., Volume 105 (1989) no. 2, pp. 307-309 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Résolution de certains modules instables et fonction de partition de Minc

Dang Ho Hai Nguyen; Lionel Schwartz; Ngoc Nam Tran

C. R. Math (2009)

Division of the Dickson algebra by the Steinberg unstable module

Nguyen Dang Ho Hai

C. R. Math (2013)

Connectivity of pseudomanifold graphs from an algebraic point of view

Karim A. Adiprasito; Afshin Goodarzi; Matteo Varbaro

C. R. Math (2015)