We show that the class of inductive limits of the representations of finite symmetric groups with simple spectrum coinsides with the class of Markov representations of the infinite symmetric group associated with Markov measures on the space of infinite Young tableaux.
We also show that the representations of infinite symmetric group induced from identity representation of two-block Young subgroup are Markov representations and find explicit formulas for transition probabilities of corresponding Markov measure on the Young diagrmas.
Induced two-row representations of finite symmetric group are studied using tensor model of those representations which alows easily to obtain the formulas for Gel'fand-Zetlin basis.
Based on the direct-sum-decomposition of the rth tensor power of the defining representation of the special orthogonal group SO(2k+1) one is interested in a bijective approach for determining the Frobenius characters of the isotypic components. In particular this leads us to a bijection between vacillating tableaux and pairs of standard Young tableaux and orthogonal Littlewood-Richardson tableaux, which we present for SO(3). Moreover we introduce the descent set of a vacillating tableau. As our bijection preserves this descent set, we also obtain the quasi-symmetric expansion of the Frobenius …
We present a bijection between vacillating tableaux and pairs consisting of a standard Young tableau and an orthogonal Littlewood-Richardson tableau for the special orthogonal group $\mathrm{SO}(2k+1)$. This bijection is motivated by the direct-sum-decomposition of the $r$th tensor power of the defining representation of $\mathrm{SO}(2k+1)$. To formulate it, we use Kwon's orthogonal Littlewood-Richardson tableaux and introduce new alternative tableaux they are in bijection with. Moreover we use a suitably defined descent set for vacillating tableaux to determine the quasi-symmetric expansion o…
For $\ell \geq 1$ and $k \geq 2$, we consider certain admissible sequences of $k-1$ lattice paths in a colored $\ell \times \ell$ square. We show that the number of such admissible sequences of lattice paths is given by the sum of squares of the number of standard Young tableaux of partitions of $\ell$ with height $\leq k$, which is also the number of $(k+1)k\cdots21$-avoiding permutations of $\{1, 2, \ldots, \ell\}$. Finally, we apply this result to the representation theory of the affine Lie algebra $\widehat{sl}(n)$ and show that this quantity gives the multiplicity of certain maximal domin…
The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine Kac--Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the {\em $($spin$)$ rigid tableaux}, and prove that they are equinumerous to the w…
We introduce and study some affine Hecke algebras of type ADE, generalising the affine Hecke algebras of GL. We construct irreducible calibrated representations and describe the calibrated spectrum. This is done in terms of new families of combinatorial objects equipped with actions of the corresponding Weyl groups. These objects are built from and generalise the usual standard Young tableaux, and are controlled by the considered affine Hecke algebras. By restriction and limiting procedure, we obtain several combinatorial models for representations of finite Hecke algebras and Weyl groups of t…
We consider the decomposition into irreducible components of the external power $Λ^p(\mathbb{C}^m\otimes \mathbb{C}^n)$ regarded as a $\operatorname{GL}_m\times\operatorname{GL}_n$-module. Skew Howe duality implies that the Young diagrams from each pair $(λ,μ)$ which contributes to this decomposition turn out to be conjugate to each other, i.e.~$μ=λ'$. We show that the Young diagram $λ$ which corresponds to a randomly selected irreducible component $(λ,λ')$ has the same distribution as the Young diagram which consists of the boxes with entries $\leq p$ of a random Young tableau of rectangular …
We study an infinite version of the "jeu de taquin" sliding game, which can be thought of as a natural measure-preserving transformation on the set of infinite Young tableaux equipped with the Plancherel probability measure. We use methods from representation theory to show that the Robinson-Schensted-Knuth ($\operatorname {RSK}$) algorithm gives an isomorphism between this measure-preserving dynamical system and the one-sided shift dynamics on a sequence of independent and identically distributed random variables distributed uniformly on the unit interval. We also show that the jeu de taquin …
In this paper, we study a new cyclic sieving phenomenon on the set $\mathsf{SST}_n(λ)$ of semistandard Young tableaux with the cyclic action $\mathsf{c}$ arising from its $U_q(\mathfrak{sl}_n)$-crystal structure. We prove that if $λ$ is a Young diagram with $\ell(λ) < n$ and $\gcd( n, |λ| )=1$, then the triple $\left( \mathsf{SST}_n(λ), \mathsf{C}, q^{- κ(λ)} s_λ(1,q, \ldots, q^{n-1}) \right) $ exhibits the cyclic sieving phenomenon, where $\mathsf{C}$ is the cyclic group generated by $\mathsf{c}$. We further investigate a connection between $\mathsf{c}$ and the promotion $\mathsf{pr}$ and sho…
The details of Lagrangian description of irreducible integer higher-spin representations of the Poincare group with an Young tableaux $Y[\hat{s}_1,\hat{s}_2]$ having $2$ columns are considered for Bose particles propagated on an arbitrary dimensional Minkowski space-time. The procedure is based, first, on using of an auxiliary Fock space generated by Fermi oscillators (antisymmetric basis), second, on construction of the Verma module and finding auxiliary oscillator realization for $sl(2)\oplus sl(2)$ algebra which encodes the second-class operator constraints subsystem in the HS symmetry supe…
