The mod 2 Margolis homology of the Dickson algebra

Hưng, Nguyễn H. V.

Comptes Rendus. Mathématique, Tome 358 (2020) no. 4, pp. 505-510.

Dans cette note on calcule entièrement l’homologie de Margolis modulo 2 de l’algèbre de Dickson $D_n$, i.e. l’homologie de $D_n$ en choisissant pour différentielles les opérations de Milnor $Q_j$, pour tous $n$ et $j$. La motivation pour cette étude est le rôle clé joué par cette homologie dans l’étude de la K-théorie de Morava $K{\left(j\right)}^{*}\left(B{S}_m\right)$ du groupe symétrique $S_m$ en $m$ lettres.

Nous montrons que la conjecture de Pengelley–Sinha sur $H_{*}(D_n;Q_j)$ pour $n\le j$ est vraie si et seulement si $n=1,2$. Pour $3\le n\le j$ notre résultat montre que la conjecture est fausse à cause de l’occurence d’éléments « critiques » $h_{s_1,\cdots ,s_k}$ de degré $(2^{j+1}-2^n)+{\sum }_{i=1}^k(2^n-2^{s_i})$ dans cette homologie pour $0<s_1<\cdots <s_k<n$ et $k>1$.

We completely compute the mod 2 Margolis homology of the Dickson algebra $D_n$, i.e. the homology of $D_n$ with the differential to be the Milnor operation $Q_j$, for every $n$ and $j$. The motivation for this problem is that, the Margolis homology of the Dickson algebra plays a key role in study of the Morava K-theory $K{\left(j\right)}^{*}\left(B{S}_m\right)$ of the symmetric group on $m$ letters $S_m$.

We show that Pengelley–Sinha’s conjecture on $H_{*}(D_n;Q_j)$ for $n\le j$ is true if and only if $n=1$ or $2$. For $3\le n\le j$, our result proves that this conjecture turns out to be false since the occurrence of some “critical elements” $h_{s_1,\cdots ,s_k}$’s of degree $(2^{j+1}-2^n)+{\sum }_{i=1}^k(2^n-2^{s_i})$ in this homology for $0<s_1<\cdots <s_k<n$ and $k>1$.

Classification : 55S05, 55S10, 55N99

[1] Dickson, Leonard E. A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Am. Math. Soc., Volume 12 (1911), pp. 75-98 | DOI | MR | Zbl

[2] Hưng, Nguyễn H. V. The action of the Steenrod squares on the modular invariants of linear groups, Proc. Am. Math. Soc., Volume 113 (1991) no. 4, pp. 1097-1104 | DOI | MR | Zbl

[3] Hưng, Nguyễn H. V.; Peterson, Franklin P. Spherical classes and the Dickson algebra, Proc. Camb. Philos. Soc., Volume 124 (1998) no. 2, pp. 253-264 | MR | Zbl

[4] Milnor, John W. The Steenrod algebra and its dual, Ann. Math., Volume 67 (1958), pp. 150-171 | DOI | MR | Zbl

[5] Mùi, Huỳnh Modular invariant theory and the cohomology algebras of symmetric group, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 22 (1975), pp. 319-369 | MR | Zbl

[6] Sinha, D. P. Cohomology of symmetric groups, 2019 (Lecture on the Vietnam-US Mathematical joint Metting, Quynhon June 10-13)

[7] Smith, Larry; Switzer, Robert M. Realizability and nonrealizability of Dickson algebras as cohomology rings, Proc. Am. Math. Soc., Volume 89 (1983), pp. 303-313 | DOI | MR | Zbl

[8] Sum, Nguyễn The action of the primitive Steenrod-Milnor operations on the modular invariants, Proceedings of the school and conference in algebraic topology, the Vietnam National University, Hanoi, Vietnam, August 9–20, 2004 (Geometry and Topology Monographs), Volume 11, Geometry & Topology Publications, 2004, pp. 349-367 | Zbl

[9] Wilkerson, Clarence A primer on the Dickson invariants, Proceedings of the Northwestern homotopy theory conference (Contemporary Mathematics), Volume 19, American Mathematical Society, 1983, pp. 421-434 | DOI | MR | Zbl

[10] Yagita, Nobuaki On the Steenrod algebra of Morava K-theory, $K$-Theory, Volume 22 (1980), pp. 423-438 | MR | Zbl