Polynomiality of shifted Plancherel averages and content evaluations

The shifted Plancherel measure is a natural probability measure on strict partitions. We prove a polynomiality property for the averages of the shifted Plancherel measure. As an application, we give alternative proofs of some content evaluation formulas, obtained by Han and Xiong very recently. Our main tool is factorial Schur $Q$-functions.Comment: 23 page

Hyperdeterminantal expressions for Jack functions of rectangular shapes

We derive a Jacobi-Trudi type formula for Jack functions of rectangular shapes. In this formula, we make use of a hyperdeterminant, which is Cayley's simple generalization of the determinant. In addition, after developing the general theory of hyperdeterminants, we give summation formulae for Schur functions involving hyperdeterminants, and evaluate Toeplitz-type hyperdeterminants by using Jack function theory.Comment: 18 pages. The title of 3rd version is ''Hyperdeterminant expressions ...''. The title of 1st and 2nd versions was ``Summation formulas for Schur functions involving hyperdeterminants'

ABJ Fractional Brane from ABJM Wilson Loop

We present a new Fermi gas formalism for the ABJ matrix model. This formulation identifies the effect of the fractional M2-brane in the ABJ matrix model as that of a composite Wilson loop operator in the corresponding ABJM matrix model. Using this formalism, we study the phase part of the ABJ partition function numerically and find a simple expression for it. We further compute a few exact values of the partition function at some coupling constants. Fitting these exact values against the expected form of the grand potential, we can determine the grand potential with exact coefficients. The results at various coupling constants enable us to conjecture an explicit form of the grand potential for general coupling constants. The part of the conjectured grand potential from the perturbative sum, worldsheet instantons and bound states is regarded as a natural generalization of that in the ABJM matrix model, though the membrane instanton part contains a new contribution.Comment: 28 pages, 5 eps figures, v3: typos corrected and references added, version to appear in JHE

Linear versus spin: representation theory of the symmetric groups

We relate the linear asymptotic representation theory of the symmetric groups to its spin counterpart. In particular, we give explicit formulas which express the normalized irreducible spin characters evaluated on a strict partition $\xi$ with analogous normalized linear characters evaluated on the double partition $D(\xi)$. We also relate some natural filtration on the usual (linear) Kerov-Olshanski algebra of polynomial functions on the set of Young diagrams with its spin counterpart. Finally, we give a spin counterpart to Stanley formula for the characters of the symmetric groups.Comment: 41 pages. Version 2: new text about non-oriented (but orientable) map