Related papers: Young tableaux and the Steenrod algebra

We denote by $\mathbb Z_2$ the prime field of two elements and by $P_t = \mathbb Z_2[x_1, \ldots, x_t]$ the polynomial algebra of $t$ generators $x_1, \ldots, x_t$ with the degree of each $x_i$ being one. Let $\mathcal A_2$ be the Steenrod…

A new tropical plactic algebra is introduced in which the Knuth relations are inferred from the underlying semiring arithmetics, encapsulating the ubiquitous plactic monoid $\mathcal{P}_n$. This algebra manifests a natural framework for…

The main result of this paper is an application of the topology of the space $Q(X)$ to obtain results for the cohomology of the symmetric group on $d$ letters, $\Sigma_d$, with `twisted' coefficients in various choices of Young modules and…

Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ with the degree of each generator $x_i$ being 1, where $\mathbb F_2$ denote the prime field with two elements. The hit problem of Frank Peterson asks for a…

We write $\mathbb P$ for the polynomial algebra in one variable over the finite field $\mathbb Z_2$ and $\mathbb P^{\otimes t} = \mathbb Z_2[x_1, \ldots, x_t]$ for its $t$-fold tensor product with itself. We grade $\mathbb P^{\otimes t}$ by…

This paper provides analogues of the results of [G.Walker and R.M.W. Wood, Linking first occurrence polynomials over F_2 by Steenrod operations, J. Algebra 246 (2001), 739--760] for odd primes p. It is proved that for certain irreducible…

Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ over the prime field of two elements, $\mathbb F_2$, with the degree of each $x_i$ being 1. We study the hit problem, set up by Frank Peterson, of finding a…

We study the category of left unital graded modules over the Steinberg algebra of a graded ample Hausdorff groupoid. In the first part of the paper, we show that this category is isomorphic to the category of unital left modules over the…

Let $A$ denote the Steenrod algebra at the prime 2 and let $k = \mathbb Z_2.$ An open problem of homotopy theory is to determine a minimal set of $A$-generators for the polynomial ring $P_q = k[x_1, \ldots, x_q] = H^{*}(k^{q}, k)$ on $q$…

Let $p$ be an odd prime number. Denote by $GL_n = GL(n,\mathbb F_p)$ the general linear group over the prime field $\mathbb F_p$. Each subgroup of $GL_n$ acts on the algebra $P_n=E(x_1,\ldots,x_n)\otimes \mathbb F_p(y_1,\ldots,y_n)$ in the…

This paper investigates Singer's conjecture by examining the cohit module $\mathbb F_2\otimes_{\mathcal A}P^{\otimes h}$ for specific degrees and values of $h$. Utilizing hit problem techniques, we extend previous work by Mothebe et al. and…

We establish connectedness criteria for graphs associated to monomials in certain quotients of the mod 2 dual Steenrod algebra. We also investigate questions about trees and Hamilton cycles in the context of these graphs. Finally, we…

Fix $\mathbb Z/2$ is the prime field of two elements and write $\mathcal A_2$ for the mod $2$ Steenrod algebra. Denote by $GL_d:= GL(d, \mathbb Z/2)$ the general linear group of rank $d$ over $\mathbb Z/2$ and by $\mathscr P_d$ the…

The Steinberg tensor product theorem is a fundamental result in the modular representation theory of reductive algebraic groups. It describes any finite-dimensional simple module of highest weight $\lambda$ over such a group as the tensor…

Let G be a Lie group, $g = Lie(G)$ - its Lie algebra, $g*$ - the dual vector space and $\widehat G$ - the set of equivalence classes of unitary irreducible representations of $G$. The orbit method [1] establishes a correspondence between…

Consider an $n\times k$ matrix of i.i.d. Bernoulli random numbers with $p=1/2$. Dual RSK algorithm gives a bijection of this matrix to a pair of Young tableaux of conjugate shape, which is manifestation of skew Howe $GL_{n}\times…

The diamond cone is a combinatorial description for a basis of an indecomposable module for the nilpotent factor $\mathfrak n$ of a semi simple Lie algebra. After N. J. Wildberger who introduced this notion, this description was achevied…

We examine the dual of the so-called "hit problem", the latter being the problem of determining a minimal generating set for the cohomology of products of infinite projective spaces as module over the Steenrod Algebra $\mathcal{A}$ at the…

We extend Wood's graph theoretic interpretation of certain quotients of the mod $2$ dual Steenrod algebra to quotients of the mod $p$ dual Steenrod algebra where $p$ is an odd prime and to quotients of the $C_2$-equivariant dual Steenrod…

It is well-known that the Steenrod algebra $A$ is self-injective as a graded ring. We make the observation that simply changing the grading on $A$ can make it cease to be self-injective. We see also that $A$ is not self-injective as an…