Expand all Collapse all | Results 1 - 5 of 5 |
1. CMB 2011 (vol 55 pp. 462)
Hook-content Formulae for Symplectic and Orthogonal Tableaux By considering the specialisation
$s_{\lambda}(1,q,q^2,\dots,q^{n-1})$ of
the Schur function, Stanley was able to describe a formula for the
number of semistandard Young tableaux of shape $\lambda$ in terms of
the contents and hook lengths of the boxes in the Young diagram.
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 $\lambda$-tableaux.
Keywords:symplectic tableaux, orthogonal tableaux, Schur function Categories:05E05, 05E10 |
2. CMB 2011 (vol 54 pp. 255)
On an Identity due to Bump and Diaconis, and Tracy and Widom
A classical question for a Toeplitz matrix with given symbol is how to
compute asymptotics for the determinants of its reductions to finite
rank. One can also consider how those asymptotics are affected when
shifting an initial set of rows and columns (or, equivalently,
asymptotics of their minors). Bump and Diaconis
obtained a formula for such shifts involving Laguerre polynomials and
sums over symmetric groups. They also showed how the Heine identity
extends for such minors, which makes this question relevant to Random
Matrix Theory. Independently, Tracy and Widom
used the Wiener-Hopf factorization to
express those shifts in terms of products of infinite matrices. We
show directly why those two expressions are equal and uncover some
structure in both formulas that was unknown to their authors. We
introduce a mysterious differential operator on symmetric functions
that is very similar to vertex operators. We show that the
Bump-Diaconis-Tracy-Widom identity is a differentiated version of the
classical Jacobi-Trudi identity.
Keywords:Toeplitz matrices, Jacobi-Trudi identity, SzegÅ limit theorem, Heine identity, Wiener-Hopf factorization Categories:47B35, 05E05, 20G05 |
3. CMB 2008 (vol 51 pp. 584)
On Tensor Products of Polynomial Representations We determine the necessary and sufficient combinatorial
conditions for which the tensor product of two irreducible polynomial
representations of $\GL(n,\mathbb{C})$ is isomorphic to another.
As a consequence we discover families of Littlewood--Richardson
coefficients that are non-zero, and a condition on Schur non-negativity.
Keywords:polynomial representation, symmetric function, Littlewood--Richardson coefficient, Schur non-negative Categories:05E05, 05E10, 20C30 |
4. CMB 2008 (vol 51 pp. 424)
Noncommutative Symmetric Bessel Functions The consideration of tensor products of $0$-Hecke algebra modules
leads to natural analogs of the Bessel $J$-functions in the algebra
of noncommutative symmetric functions. This provides a simple explanation
of various combinatorial properties of Bessel functions.
Categories:05E05, 16W30, 05A15 |
5. CMB 2006 (vol 49 pp. 281)
Correction to a Theorem on Total Positivity A well-known theorem states that if $f(z)$ generates a PF$_r$
sequence then $1/f(-z)$ generates a PF$_r$ sequence. We give two
counterexamples
which show that this is not true, and give a correct version of the theorem.
In the infinite limit the result is sound: if $f(z)$ generates a PF
sequence then $1/f(-z)$ generates a PF sequence.
Keywords:total positivity, Toeplitz matrix, PÃ³lya frequency sequence, skew Schur function Categories:15A48, 15A45, 15A57, 05E05 |