Narrow Results

Let $G$ be a Hermitian-type Lie group with the complexified Lie algebra $𝔤$. We use $L(\lambda )$ to denote a highest weight Harish-Chandra $G$-module with infinitesimal character $\lambda$. Let $w$ be an element in the Weyl group $W$. We use $L_w$ to denote a highest weight module with highest weight $-w\rho -\rho$. In this paper we prove that there is only one Kazhdan–Lusztig right cell such that the corresponding highest weight Harish-Chandra modules $L_w$ have the same associated variety. Then we give a characterization for those $w$ such that $L_w$ is a highest weight Harish-Chandra module and the associated variety of $L(\lambda )$ will be characterized by the information of the Kazhdan–Lusztig right cell containing some special $w_{\lambda }$. We also count the number of those highest weight Harish-Chandra modules $L_w$ in a given Harish-Chandra cell.

We construct the analog of the plactic monoid for the super semistandard Young tableaux over a signed alphabet. This is done by developing a generalization of the Knuth's relations. Moreover we get generalizations of Greene's invariants and Young–Pieri rule. A generalization of the symmetry theorem in the signed case is also obtained. Except for this last result, all the other results are proved without restrictions on the orderings of the alphabets.

In this paper, we present the results of a computer investigation of asymptotics for maximum dimensions of linear and projective representations of the symmetric group. This problem reduces the investigation of standard and strict Young diagrams of maximum dimensions. We constructed some sequences for both standard and strict Young diagrams with extremely large dimensions. The conjecture that the limit of normalized dimensions exists was proposed by Vershik 30 years ago [A. M. Vershik and S. V. Kerov, Funktsional Anal. i Prilozhen19(1) (1985) 25–36] and has not been proved yet. We studied the growth and oscillations of the normalized dimension function in sequences of Young diagrams. Our approach is based on analyzing finite differences of their normalized dimensions. This analysis also allows us to give much more precise estimation of the limit constants.

We prove that the quantized Carter–Lusztig basis for a finite-dimensional irreducible U_q(𝔤𝔩_n(ℂ))-module V(λ) is related to the global crystal basis for V(λ) by an upper triangular invertible matrix. We express the global crystal basis in terms of the q-Schur algebra and provide an algorithm for obtaining global crystal basis vectors for V(λ) using the q-Schur algebra.

Consider the affine Lie algebra $\widehat{s\ell }(n)$ with null root $\delta$, weight lattice $P$ and set of dominant weights $P^{+}$. Let $V(k{\mathrm{\Lambda }}_0),$$k\in {\mathbb{Z}}_{\ge 1}$ denote the integrable highest weight $\widehat{s\ell }(n)$-module with level $k\ge 1$ highest weight $k{\mathrm{\Lambda }}_0$. Let $\mathrm{\text{wt}}(V)$ denote the set of weights of $V(k{\mathrm{\Lambda }}_0)$. A weight $\mu \in \mathrm{\text{wt}}(V)$ is a maximal weight if $\mu +\delta \notin \mathrm{\text{wt}}(V)$. Let ${\mathrm{\text{max}}}^{+}(k{\mathrm{\Lambda }}_0)=\mathrm{\text{max}}(k{\mathrm{\Lambda }}_0)\cap P^{+}$ denote the set of maximal dominant weights which is known to be a finite set. The explicit description of the weights in the set ${\mathrm{\text{max}}}^{+}(k{\mathrm{\Lambda }}_0)$ is known [R. L. Jayne and K. C. Misra, On multiplicities of maximal dominant weights of $\widehat{sl}(n)$-modules, Algebr. Represent. Theory17 (2014) 1303–1321]. In papers [R. L. Jayne and K. C. Misra, Lattice paths, Young tableaux, and weight multiplicities, Ann. Comb.22 (2018) 147–156; R. L. Jayne and K. C. Misra, Multiplicities of some maximal dominant weights of the $\widehat{s\ell }(n)$-modules $V(k{\mathrm{\Lambda }}_0)$, Algebr. Represent. Theory25 (2022) 477–490], the multiplicities of certain subsets of ${\mathrm{\text{max}}}^{+}(k{\mathrm{\Lambda }}_0)$ were given in terms of some pattern-avoiding permutations using the associated crystal base theory. In this paper the multiplicity of all the maximal dominant weights of the $\widehat{s\ell }(n)$-module $V(k{\mathrm{\Lambda }}_0)$ are given generalizing the results in [R. L. Jayne and K. C. Misra, Lattice paths, Young tableaux, and weight multiplicities, Ann. Comb.22 (2018) 147–156; R. L. Jayne and K. C. Misra, Multiplicities of some maximal dominant weights of the $\widehat{s\ell }(n)$-modules $V(k{\mathrm{\Lambda }}_0)$, Algebr. Represent. Theory25 (2022) 477–490].

Task reassignments in two-dimensional (2D) mesh-connected systems (2D-MSs) have been researched for several decades. We propose a hierarchical 2D mesh-connected system (2D-HMS) in order to exploit the regular nature of a 2D-MS. In our approach priority-based task assignments and reassignments in a 2D-HMS are represented by tableaux and their algorithms. We show how task relocations for a priority-based task reassignment in a 2D-HMS are reduced to a jeu de taquin slide.