A Note on the Algebra of Operations for Hopf Cohomology at Odd Primes

Let $p$ be any prime, and let ${\mathcal B}(p)$ be the algebra of operations on the cohomology ring of any cocommutative $\mathbb{F}_p$-Hopf algebra. In this paper we show that when $p$ is odd (and unlike the $p=2$ case), ${\mathcal B}(p)$ cannot become an object in the Singer category of $\mathbb{F}_p$-algebras with coproducts, if we require that coproducts act on the generators of ${\mathcal B}(p)$ coherently with their nature of cohomology operation

Signed bicyclic graphs with minimal index

The index of a signed graph \Sigma = (G; \sigma) is just the largest eigenvalue of its adjacency matrix. For any n > 4 we identify the signed graphs achieving the minimum index in the class of signed bicyclic graphs with n vertices. Apart from the n = 4 case, such graphs are obtained by considering a starlike tree with four branches of suitable length (i.e. four distinct paths joined at their end vertex u) with two additional negative independent edges pairwise joining the four vertices adjacent to u. As a by-product, all signed bicyclic graphs containing a theta-graph and whose index is less than 2 are detected

On the multiplicity of α as an A_α (Γ)-eigenvalue of signed graphs with pendant vertices

A signed graph is a pair Γ = (G; ), where x = (V (G);E(G)) is a graph and E(G) -> {+1;−1} is the sign function on the edges of G. For any > [0; 1] we consider the matrix Aα(Γ) = αD(G) + (1 −α )A(Γ); where D(G) is the diagonal matrix of the vertex degrees of G, and A(Γ) is the adjacency matrix of Γ. Let mAα(Γ) be the multiplicity of α as an A(Γ)-eigenvalue, and let G have p(G) pendant vertices, q(G) quasi-pendant vertices, and no components isomorphic to K2. It is proved that mA(Γ)() = p(G) − q(G) whenever all internal vertices are quasi-pendant. If this is not the case, it turns out that mA(Γ)() = p(G) − q(G) +mN(Γ)(); where mN(Γ)() denotes the multiplicity of as an eigenvalue of the matrix N(Γ) obtained from A(Γ) taking the entries corresponding to the internal vertices which are not quasipendant. Such results allow to state a formula for the multiplicity of 1 as an eigenvalue of the Laplacian matrix L(Γ) = D(G) − A(Γ). Furthermore, it is detected a class G of signed graphs whose nullity – i.e. the multiplicity of 0 as an A(Γ)-eigenvalue – does not depend on the chosen signature. The class G contains, among others, all signed trees and all signed lollipop graphs. It also turns out that for signed graphs belonging to a subclass G ` G the multiplicity of 1 as Laplacian eigenvalue does not depend on the chosen signatures. Such subclass contains trees and circular caterpillars

Length-preserving monomorphisms for Steenrod algebras at odd primes

Let p be an odd prime. In this paper we determine the group of length-preserving automorphisms for the ordinary Steenrod algebra A(p) and for B(p), the algebra of cohomology operations for the cohomology of cocom- mutative Fp-Hopf algebras. Contrarily to the p = 2 case, no length-preserving strict monomorphism turns out to exis

An algebraic introduction to the Steenrod algebra

This paper is a complete presentation of the Steenrod algebra P from a purely algebraic point of view, without referring to cohomology operations. Let Fq be a Galois field of characteristic p, V a finite-dimensional vector space over Fq and Fq[V ] the graded algebra of polynomial functions on V . Fq[−] is a contravariant functor from Fq-vector spaces to graded connected algebras. Then P can be defined as the subalgebra of the endomorphism algebra of Fq[−] generated by the homogeneous components of a perturbation of the Frobenius map [L. Smith, Polynomial invariants of finite groups, A K Peters, Wellesley, MA, 1995; MR1328644 (96f:13008)]. The generators of P obey the Adem-Wu relations; they can be derived by the Bullett-Macdonald identity [S. R. Bullett and I. G. Macdonald, Topology 21 (1982), no. 3, 329–332; MR0649764 (83h:55035)]. The proofs to show that the Adem-Wu relations are a complete set of defining relations for P are rearranged following the strategy of H. Cartan [Comment. Math. Helv. 29 (1955), 40–58; MR0068219 (16,847e)] and J.-P. Serre [Comment. Math. Helv. 27 (1953), 198–232; MR0060234 (15,643c)]. They are extended from the prime field case to the case of an arbitrary Galois field Fq, q = p . The paper also includes the Hopf algebra structure of P [J. Milnor, Ann. of Math. (2) 67 (1958), 150–171; MR0099653 (20 #6092)] with Milnor’s theorems extended from Fp to Fq. {For the entire collection see MR2404071 (2009a:55001)

On triviality of Dickson invariants in the homology of the Steenrod algebra

Let A be the Steenrod algebra and let Dk denote the Dickson algebra of k variables. J. E. Lannes and S. Zarati [Math. Z. 194 (1987), no. 1, 25–59; MR0871217 (88j:55014)] defined homomorphisms 'k: Extk,k+i A (F2,F2) ! (F2 A Dk) i , which correspond to an associated graded of the Hurewicz homomorphism H: s (S0) = (Q0S0)!H (Q0S0;F2). The long-standing geometric conjecture, that only Hopf invariant 1 and Kervaire invariant 1 classes are detected by H, has the following algebraic version: 'k = 0 in positive stems, for k > 2. It has been proved that '3 = 0 [Nguy˜ˆen H˜u’u Vi.ˆet Hu’ng, Trans. Amer. Math. Soc. 349 (1997), no. 10, 3893–3910; MR1433119 (98e:55020)] and that 'k vanishes on the image of Singer’s algebraic transfer for k > 2 [Nguy˜ˆen H˜u’u Vi.ˆet Hu’ng and Trˆan Ngoc Nam, Trans. Amer. Math. Soc. 353 (2001), no. 12, 5029–5040 (electronic); MR1852092 (2002f:55041)]. Further, by [Nguy˜ˆen H˜u’u Vi.ˆet Hu’ng and F. P. Peterson, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 2, 253–264; MR1631123 (99i:55021)] we know that 'k vanishes on decomposable elements for k > 2 and that '4 = 0 in positive stems < 89. In the paper under review the author completes this last result, establishing the conjecture for k = 4. The key step in the proof of his result is to show that the squaring operation Sq0 on (F2 ADk) , defined in [Nguy˜ˆen H˜u’u Vi.ˆet Hu’ng, op. cit.], commutes with the classical squaring operation Sq0 on Extk A(F2,F2) through the 'k. To this end, the explicit chain-level representation of the Lannes-Zarati dual map ' k [see Nguy˜ˆen H˜u’u Vi.ˆet Hu’ng, Trans. Amer. Math. Soc. 353 (2001), no. 11, 4447–4460 (electronic); MR1851178 (2002g:55032)] plays a key role and leads to the following equivalent formulation of the conjecture taken into account: the Dickson invariants are homologically trivial in TorAk (F2,F2)

Symmetric coinvariant algebras and local Weyl modules at a double point

Let Sn be the symmetric group acting on C[x1, . . . , xn]. The classical symmetric coinvariant algebra C[x1, . . . , xn]Sn is the quotient of C[x1, . . . , xn] by the ideal generated by symmetric polynomials vanishing at (0, . . . , 0). According to a classical result, it is isomorphic to C[Sn] as Sn-module. In [Comm. Math. Phys. 251 (2004), no. 3, 427–445; MR2102326 (2005m:17005)], B. L. Fe˘ıgin and S. A. Loktev defined the symmetric coinvariant algebra A n Sn , where A is the coordinate ring of an affine variety M over C. In the paper under review the author deals with A = C[x, y]/(xy), the coordinate ring ofM = {(x, y) 2 C2: xy = 0}. In this case the symmetric coinvariant algebra is Rn = A n/Jn, where Jn is the ideal of A n generated by the elementary symmetric polynomials ei = ei(x1, . . . , xn), fi = fi(x1, . . . , xn), 1 i n. He introduces a generalization of Rn, Rn i,j = A n/In i,j , for 1 i, j n, and gives its Snmodule structure when i+j n+1. This description is then used to show that, as Sn-module, Rn = C[Sn] (n−1)IndSn S2L1,1, where IndSn S2L(1,1) = C[Sn] C[S2] L(1,1) and L(1,1) is the sign representation of S2. This result, combined with a theorem contained in the paper by Fe˘ıgin and Loktev quoted above, gives another description of the slr+1-module structure of the local Weyl module at the double point 0 ofM for slr+1 A

Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups

Let F be a field of characteristic p and let F[V ] denote the symmetric algebra on the dual V of V = Fn. Let H be an unstable Noetherian integral domain and denote by P p H the inseparable extension of H over the Steenrod algebra P of reduced powers. In [Mem. Amer. Math. Soc. 146 (2000), no. 692, x+158 pp.; MR1693799 (2000m:55023)] M. D. Neusel showed that H can be embedded into F[V ] and P p H = F[V ]G, for some finite subgroup G of GL(V ). In the paper under review the author extends these results and those in [C. W. Wilkerson, Jr., in Recent progress in homotopy theory (Baltimore, MD, 2000), 381–396, Contemp. Math., 293, Amer. Math. Soc., Providence, RI, 2002; MR1890745 (2003d:55020)], proving that, if the extension H ,!F[V ]G is of exponent e, then V decomposes as a direct sum of subspaces W0,W1, . . . ,We such that G acts on the flags W0 W1 · · · Wi, for i = 0, . . . , e. Further, the integral closure H of H is equal to the G-invariant subalgebra (F[W0] F[W1]p · · · F[We]pe)G, and the field of fractions H of H is equal to (F(W0) F(W1)p · · · F(We)pe)G. It turns out that G consists of flag matrices whose form depends on the vector space dimensions of the Wi’s. Neusel also proves the following results: H is Cohen-Macaulay if and only if P p H is Cohen-Macaulay; H is polynomial if and only if P p H is polynomial and the generators are pth powers/roots of one another. Combining these results, a twenty-year-old conjecture formulated byWilkerson is settled (see Conjecture 5.1 in [C.W. Wilkerson, Jr., op. cit.])

On a theorem of R. Steinberg on rings of coinvariants

Let :G ,!GL(n, F) be a representation of a finite group G over the field F. G acts via on the ring of polynomial functions F[V ], V = Fn. LetH(G) F[V ] be the Hilbert ideal, i.e., the ideal in F[V ] generated by all the homogeneous G-invariant forms of strictly positive degree. R. Steinberg [Trans. Amer. Math. Soc. 112 (1964), 392–400; MR0167535 (29 #4807)] proved that for F of characteristic zero, the algebra of coinvariants F[V ]G = F[V ]/H(G) satisfies Poincar´e duality if and only if G is a pseudoreflection group (formulation of R. M. Kane [Canad. Math. Bull. 37 (1994), no. 1, 82–88; MR1261561 (96e:51016)]). In the paper under review, the author explores the situation for fields of non-zero characteristic. He proves that an analogue of Steinberg’s theorem holds for the case n = 2 and, for n = 3, it holds in a weak sense for representations of p-groups over a field F of characteristic p. He gives a counterexample in the case n = 4, using the vector invariants of Z/2 over F2: F2[V ]Z/2 is a Poincar´e duality algebra, although F2[V ]Z/2 is not a polynomial algebra and Z/2 contains no pseudoreflections. He also shows the following interesting property: F2[V ]Z/2 is isomorphic to the algebra of coinvariants obtained from a degree 4 faithful representation of the group Z/2× Z/2, which is generated by reflections and has polynomial invariants