{"title":"Steenrod operations on the modular invariants","authors":"Nguyễn Sum","doi":"10.2996/kmj/1138040053","DOIUrl":null,"url":null,"abstract":"In this paper, we compute the action of the mod p Steenrod operations on the modular invariants of the linear groups with p an odd prime number. Introduction Fix an odd prime p. Let Apn be the alternating group on p letters. Denote by Σpn,p a Sylow p-subgroup of Apn and E an elementary abelian p-group of rank n. Then we have the restriction homomorphisms Res(E, Σpn,p) : H (BΣpn,p) −→ H (BE), Res(E, Apn) : H (BApn ) −→ H (BE), induced by the regular permutation representation E ⊂ Σpn,p ⊂ Apn of E (see Mùi [4]). Here and throughout the paper, we assume that the coefficients are taken in the prime field Z/p. Using modular invariant theory of linear groups, Mùi proved in [3, 4] that ImRes(E, Σpn,p) = E(U1, . . . , Un) ⊗ P (V1, . . . , Vn), ImRes(E, Apn ) = E(M̃n,0, . . . , M̃n,n−1) ⊗ P (L̃n, Qn,1, . . . , Qn,n−1), Here and in what follows, E(., . . . , .) and P (., . . . , .) are the exterior and polynomial algebras over Z/p generated by the variables indicated. L̃n, Q,s are the Dickson invariants of dimensions p, 2(p − p), and M̃n,s, , Uk, Vk are the Mùi invariants of dimensions p − 2p, pk−1, 2pk−1 respectively (see Section 1). Let A be the mod p Steenrod algebra and let τs, ξi be the Milnor elements of dimensions 2p − 1, 2p − 2 respectively in the dual algebra A∗ of A. In [7], Milnor showed that, as an algebra A∗ = E(τ0, τ1, . . .) ⊗ P (ξ1, ξ2, . . .). Then A∗ has a basis consisting of all monomials τSξ = τs0 . . . τsk ξ r1 . . . ξm , with S = (s1, . . . , sk), 0 6 s1 < . . . < sk, R = (r1, . . . , rm), ri > 0. Let St ∈ A denote the dual of τSξ with respect to that basis. Then A has a basis consisting all operations St. For S = ∅, R = (r), St∅,(r) is nothing but the Steenrod operation P . Since H(BG), G = E, Σpn,p or Apn , is an A-module (see [13, Chap. VI]) and the restriction homomorphisms are A-linear, their images are A-submodules of H(BE). 2010 Mathematics Subject Classification. Primary 55S10; Secondary 55S05.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2021-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Kodai Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2996/kmj/1138040053","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 8
Abstract
In this paper, we compute the action of the mod p Steenrod operations on the modular invariants of the linear groups with p an odd prime number. Introduction Fix an odd prime p. Let Apn be the alternating group on p letters. Denote by Σpn,p a Sylow p-subgroup of Apn and E an elementary abelian p-group of rank n. Then we have the restriction homomorphisms Res(E, Σpn,p) : H (BΣpn,p) −→ H (BE), Res(E, Apn) : H (BApn ) −→ H (BE), induced by the regular permutation representation E ⊂ Σpn,p ⊂ Apn of E (see Mùi [4]). Here and throughout the paper, we assume that the coefficients are taken in the prime field Z/p. Using modular invariant theory of linear groups, Mùi proved in [3, 4] that ImRes(E, Σpn,p) = E(U1, . . . , Un) ⊗ P (V1, . . . , Vn), ImRes(E, Apn ) = E(M̃n,0, . . . , M̃n,n−1) ⊗ P (L̃n, Qn,1, . . . , Qn,n−1), Here and in what follows, E(., . . . , .) and P (., . . . , .) are the exterior and polynomial algebras over Z/p generated by the variables indicated. L̃n, Q,s are the Dickson invariants of dimensions p, 2(p − p), and M̃n,s, , Uk, Vk are the Mùi invariants of dimensions p − 2p, pk−1, 2pk−1 respectively (see Section 1). Let A be the mod p Steenrod algebra and let τs, ξi be the Milnor elements of dimensions 2p − 1, 2p − 2 respectively in the dual algebra A∗ of A. In [7], Milnor showed that, as an algebra A∗ = E(τ0, τ1, . . .) ⊗ P (ξ1, ξ2, . . .). Then A∗ has a basis consisting of all monomials τSξ = τs0 . . . τsk ξ r1 . . . ξm , with S = (s1, . . . , sk), 0 6 s1 < . . . < sk, R = (r1, . . . , rm), ri > 0. Let St ∈ A denote the dual of τSξ with respect to that basis. Then A has a basis consisting all operations St. For S = ∅, R = (r), St∅,(r) is nothing but the Steenrod operation P . Since H(BG), G = E, Σpn,p or Apn , is an A-module (see [13, Chap. VI]) and the restriction homomorphisms are A-linear, their images are A-submodules of H(BE). 2010 Mathematics Subject Classification. Primary 55S10; Secondary 55S05.
期刊介绍:
Kodai Mathematical Journal is edited by the Department of Mathematics, Tokyo Institute of Technology. The journal was issued from 1949 until 1977 as Kodai Mathematical Seminar Reports, and was renewed in 1978 under the present name. The journal is published three times yearly and includes original papers in mathematics.