Combinatorial constructions of weight bases: the Gelfand-Tsetlin basis.
Combinatorial Laplacian of the matching complex.
Combinatorial operators for Kronecker powers of representations of .
Combinatorial topology and the global dimension of algebras arising in combinatorics
In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands. In both cases, the representation theory of these monoids played a prominent role. In particular, it was used to compute the spectrum of the...
Combinatorics, superalgebras, invariant theory and representation theory.
Composition of transpositions and equality of ribbon Schur -functions.
Counting Young tableaux of bounded height.
Coxeter polynomials of Salem trees
We compute the Coxeter polynomial of a family of Salem trees, and also the limit of the spectral radii of their Coxeter transformations as the number of their vertices tends to infinity. We also prove that if z is a root of multiplicities for the Coxeter polynomials of the trees respectively, then z is a root for the Coxeter polynomial of their join, of multiplicity at least where .
Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras
We define the -restriction and -induction functors on the category of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.
Cyclic sieving for longest reduced words in the hyperoctahedral group.
Decomposition of the diagonal action of on the coinvariant space of .
Determinantal transition kernels for some interacting particles on the line
We find the transition kernels for four markovian interacting particle systems on the line, by proving that each of these kernels is intertwined with a Karlin–McGregor-type kernel. The resulting kernels all inherit the determinantal structure from the Karlin–McGregor formula, and have a similar form to Schütz’s kernel for the totally asymmetric simple exclusion process.
Determination of the structure of algebraic curvature tensors by means of Young symmetrizers.
Diamond representations of
In [6], there is a graphic description of any irreducible, finite dimensional module. This construction, called diamond representation is very simple and can be easily extended to the space of irreducible finite dimensional -modules.In the present work, we generalize this construction to . We show it is in fact a description of the reduced shape algebra, a quotient of the shape algebra of . The basis used in [6] is thus naturally parametrized with the so called quasi standard Young tableaux....
Dimensions of components of tensor products of representations of linear groups with applications to Beurling-Fourier algebras
We give universal upper bounds on the relative dimensions of isotypic components of a tensor product of representations of the linear group GL(n) and universal upper bounds on the relative dimensions of irreducible components of a tensor product of representations of the special linear group SL(n). This problem is motivated by harmonic analysis problems, and we give some applications to the theory of Beurling-Fourier algebras.
Domino Fibonacci tableaux.
Domino tableaux, functions and plethysms. (Tableaux de dominos, fonctions et plethysmes.)
Double crystals of binary and integral matrices.
Edges and tableaux. (Arêtes et tableaux.)
Entropy of Schur–Weyl measures
Relative dimensions of isotypic components of th order tensor representations of the symmetric group on letters give a Plancherel-type measure on the space of Young diagrams with cells and at most rows. It was conjectured by G. Olshanski that dimensions of isotypic components of tensor representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to this family of Plancherel-type measures in the limit when converges to a constant. The main...