32 pages. This third version of the paper contains some comments on homomorphisms between the Specht modules defined by Dipper and James and has a more rigorous proof of the result following Proposition 4.1
We study the homomorphism spaces between Specht modules for the Hecke algebras $\h$ of type $A$. We prove a cellular analogue of the kernel intersection theorem and a $q$-analogue of a theorem of Fayers and Martin and apply these results to give an algorithm which computes the homomorphism spaces $\Hom_{\h}(S^μ,S^λ)$ for certain pairs of partitions $λ$ and $μ$. We give an explicit description of the homomorphism spaces $\Hom_\h(S^μ,S^λ)$ where $\h$ is an algebra over the complex numbers, $λ=(λ_1,λ_2)$ and $μ$ is an arbitrary partition with $μ_1 \geq λ_2$.
In 1976, King defined certain tableaux model, called King tableaux in this paper, counting weight multiplicities of irreducible representation of the symplectic group $Sp(2m)$ for a given dominant weight. Since Kashiwara defined crystals, it is an open problem to provide a crystal structure on King tableaux. In this paper, we present crystal structures on King tableaux and semistandard oscillating tableaux. The semistandard oscillating tableaux naturally appear as $Q$-tableaux in the symplectic version of RSK algorithms. As an application, we discuss Littlewood-Richardson coefficients for $Sp(…
Using combinatorics of Young tableaux, we give an explicit construction of irreducible graded modules over Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^λ$ of type $A_{n}$. Our construction is compatible with crystal structure. Let ${\mathbf B}(\infty)$ and ${\mathbf B}(λ)$ be the $U_q(\slm_{n+1})$-crystal consisting of marginally large tableaux and semistandard tableaux of shape $λ$, respectively. On the other hand, let ${\mathfrak B}(\infty)$ and ${\mathfrak B}(λ)$ be the $U_q(\slm_{n+1})$-crystals consisting of isomorphism classes of irreducible graded $R$-modules a…
We present several equinumerous results between generalized oscillating tableaux and semistandard tableaux and give a representation-theoretical proof to them. As one of the key ingredients of the proof, we provide Pieri rules for the symplectic and orthogonal groups.
A new combinatorial interpretation of the Howe dual pair $(\hat{\frak{gl}}_{\infty|\infty},\frak{gl}_n)$ acting on an infinite dimensional Fock space $\frak{F}^n$ of level $n$ is presented. The character of a quasi-finite irreducible highest weight representation of $\hat{\frak{gl}}_{\infty|\infty}$ occurring in $\frak{F}^n$ is realized in terms of certain bitableaux of skew shapes. We study a general combinatorics of these bitableaux, including Robinson-Schensted-Knuth correspondence and Littlewood-Richardson rule, and then its dual relation with the rational semistandard tableaux for $\frak{…
In this paper, we give an explicit combinatorial realization of the crystal B(λ) for an irreducible highest weight U_q(q(n))-module V(λ) in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algorithms of decomposing the tensor product of q(n)-crystals. Consequently, we obtain explicit combinatorial descriptions of the shifted Littlewood-Richardson coefficients.
Let $δ=(δ_1,\ldots,δ_n)$ be a string of letters $h$ and $v$. We define a Young tableau to be $δ$-semistandard if the entries are weakly increasing along rows and columns, and the entries $i$ form a horizontal strip if $δ_i=h$ and a vertical strip if $δ_i=v$. We define $δ$-promotion on such tableaux via a modified jeu-de-taquin. The first main result is that $δ$-promotion has period $n$ on rectangular $δ$-semistandard tableaux, generalizing the results of Haiman and Rhoades for standard and semistandard tableaux. The second main result states that the set of rectangular $δ$-semistandard tableau…
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…
We define crystal operators on semistandard shifted tableaux, giving a new proof that Schur $P$-functions are Schur positive. We define a queer crystal operator to construct a connected queer crystal on semistandard shifted tableaux of a given shape, providing a new proof that products of Schur $P$-functions are Schur $P$-positive. We also give a rectification map from shifted tableaux to Young tableaux that commutes with the crystal operators and provides a dual algorithm to shifted insertion.
Suppose $μ$ is a partition of $n$ and $λ$ a composition of $n$, and let $S^μ$, $M^λ$ denote the Specht module and permutation module defined by Dipper and James for the Iwahori--Hecke algebra $\mathscr{H}_n$ of the symmetric group $\mathfrak{S}_n$. We give an explicit fast algorithm for expressing a tableau homomorphism $\hatϕ_A:S^μ\to M^λ$ as a linear combination of semistandard homomorphisms. Along the way we provide a utility result related to removing rows from tableaux.
By considering the specialisation $s_λ(1,q,q^2,...,q^{n-1})$ of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape $λ$ in terms of two properties of the boxes in the diagram for $λ$. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal $λ$-tableaux.
We define a number of new combinatorial operations on skew semistandard domino tableaux, which together with constructions introduced earlier by C. Carre and B. Leclerc, define an elegant structure on the set of these tableaux, that closely parallels the structure defined in [math.CO/9908099] for ordinary skew semistandard tableaux, that is related to the Littlewood-Richardson rule. These operations are: (1) a bijection between semistandard domino tableaux and certain pairs of ordinary tableaux of the same weight that together fill the same shape; (2) a weight preserving transformation of domi…
In this paper, we first study the shape algebra and the reduced shape algebra for the Lie superalgebra $\mathfrak{sl}(m,n)$. We define the quasistandard tableaux, their collection is the diamond cone for $\mathfrak{sl}(m,n)$, which is a combinatorial basis for the reduced shape algebra. We realize a bijection between the set of semistandard tableaux with shape $λ$ and the set of quasistandard tableaux with shape $μ\leqλ$, by using the 'super jeu de taquin' on skew semistandard tableaux. This gives the compatibility of the diamond cone with the natural stratification of the reduced shape algebr…
Let $s_ν\circ s_μ$ denote the plethystic product of the Schur functions $s_ν$ and $s_μ$. In this article we define an explicit polynomial representation corresponding to $s_ν\circ s_μ$ with basis indexed by certain `plethystic' semistandard tableaux. Using these representations we prove generalizations of four results on plethysms due to Bruns--Conca--Varbaro, Brion, Ikenmeyer and the authors. In particular, we give a sufficient condition for the multiplicity $\langle s_ν\circ s_μ, s_λ\rangle$ to be stable under insertion of new parts into $μ$ and $λ$. We also characterize all maximal and mini…
This paper introduces calibrated representations for affine Hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations. The primary technique is to provide indexing sets for controlling the weight space structure of finite dimensional modules for the affine Hecke algebra. Using these indexing sets we show that (1) irreducible calibrated representations are indexed by skew local regions, (2) the dimension of an irreducible calibrated representation is the number of chambers in the local region, (3) each irreducible calibrated representation is con…
The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, planar rook monoid, and symmetric group algebras. We give a construction of the irreducible modules of these algebras in two isomorphic ways: first, as the span of symmetric diagrams on which the algebra acts by conjugation twisted with an irreducible symmetric group representation and, second, on a basis ind…
We construct a class $Θ_{\mathscr{R}}$ of homomorphisms from a Specht module $S_{\mathbb{Z}}^λ$ to a signed permutation module $M_{\mathbb{Z}}(α|β)$ which generalises James's construction of homomorphisms whose codomain is a Young permutation module. We show that any $ϕ\in \operatorname{Hom}_{\mathbb{Z}\mathfrak{S}_{n}}\big(S_{\mathbb{Z}}^λ, M_{\mathbb{Z}}(α|β)\big)$ lies in the $\mathbb{Q}$-span of $Θ_{\text{sstd}}$, a subset of $Θ_{\mathscr{R}}$ corresponding to semistandard $λ$-tableaux of type $(α|β)$. We also study the conditions for which $Θ^{\mathbb{F}}_{\mathrm{sstd}}$ - a subset of $\…
The conjugacy classes of nilpotent $n\times n$ matrices can be parametrised by partitions $λ$ of $n$, and for a nilpotent $η$ in the class parametrised by $λ$, the variety $F_η$ of $η$-stable flags has its irreducible components parametrised by the standard Young tableaux of shape $λ$. We indicate how several algorithmic constructions defined for Young tableaux have significance in this context, thus extending Steinberg's result that the relative position of flags generically chosen in the irreducible components of $F_η$ parametrised by tableaux $P$ and $Q$, is the permutation associated to $(…
The main results of this paper are accessible with only basic linear algebra. Given an increasing sequence of dimensions, a flag in a vector space is an increasing sequence of subspaces with those dimensions. The set of all such flags (the flag manifold) can be projectively coordinatized using products of minors of a matrix. These products are indexed by tableaux on a Young diagram. A basis of "standard monomials" for the vector space generated by such projective coordinates over the entire flag manifold has long been known. A Schubert variety is a subset of flags specified by a permutation. L…
