Interpretation of a basic hypergeometric identity with Lie characters and Young tableaux

A combinatorial interpretation involving semistandard tableaux is provided for a four parameter terminating sum identity for the basic hypergeometric series 6φ5. This identity is first produced by taking the principal specialization of a tensor product identity for two GL(n) characters

Interpretation of a basic hypergeometric identity with Lie characters and Young tableaux

A combinatorial interpretation involving semistandard tableaux is provided for a four parameter terminating sum identity for the basic hypergeometric series 6φ5. This identity is first produced by taking the principal specialization of a tensor product identity for two GL(n) characters

Torus knot polynomials and susy Wilson loops

We give, using an explicit expression obtained in [V. Jones, Ann. of Math. 126, 335 (1987)], a basic hypergeometric representation of the HOMFLY polynomial of $(n,m)$ torus knots, and present a number of equivalent expressions, all related by Heine's transformations. Using this result the $(m,n)\leftrightarrow (n,m)$ symmetry and the leading polynomial at large $N$ are explicit. We show the latter to be the Wilson loop of 2d Yang-Mills theory on the plane. In addition, after taking one winding to infinity, it becomes the Wilson loop in the zero instanton sector of the 2d Yang-Mills theory, which is known to give averages of Wilson loops in $\mathcal{N}$=4 SYM theory. We also give, using matrix models, an interpretation of the HOMFLY polynomial and the corresponding Jones-Rosso representation in terms of $q$-harmonic oscillators.Comment: 17 pages, v2: More concise (published) version; typos correcte

A note on moments of derivatives of characteristic polynomials

We present a simple technique to compute moments of derivatives of unitary characteristic polynomials. The first part of the technique relies on an idea of Bump and Gamburd: it uses orthonormality of Schur functions over unitary groups to compute matrix averages of characteristic polynomials. In order to consider derivatives of those polynomials, we here need the added strength of the Generalized Binomial Theorem of Okounkov and Olshanski. This result is very natural as it provides coefficients for the Taylor expansions of Schur functions, in terms of shifted Schur functions. The answer is finally given as a sum over partitions of functions of the contents. One can also obtain alternative expressions involving hypergeometric functions of matrix arguments.Comment: 12 page