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Soichi Okada

Publications and source records attributed to Soichi Okada.

24 records · Page 2Linked to original sources
arXiv2008

Instanton calculus and chiral one-point functions in supersymmetric gauge theories

We compute topological one-point functions of the chiral operator Tr ϕ^k in the maximally confining phase of U(N) supersymmetric gauge theory. These one-point functions are polynomials in the equivariant parameter \hbar and the parameter of instanton expansion q=Λ^{2N} and are of particular interest from gauge/string theory correspondence, since they are related to the Gromov-Witten theory of P^1. Based on a combinatorial identity that gives summation formula over Young diagrams of relevant functions, we find a relation among chiral one-point functions, which recursively determines the \hbar expansion of the generating function of one-point functions. Using a result from the operator formalism of the Gromov-Witten theory, we also present a vacuum expectation value of the loop operator Tr e^{itϕ}.

hep-th↗
arXiv2005

Generalizations of Cauchy's Determinant and Schur's Pfaffian

We present several generalizations of Cauchy's determinant and Schur's Pfaffian by considering matrices whose entries involve some generalized Vandermonde determinants. Special cases of our formulae include previuos formulae due to S.Okada and T. Sundquist. As an application, we give a relation for the Littlewood--Richardson coefficients involving a rectangular partition.

math.CO↗
arXiv2004

An elliptic generalization of Schur's Pfaffian identity

We present a Pfaffian identity involving elliptic functions, whose rational limit gives a generalization of Schur's Pfaffian identity for Pf ((x_j - x_i)/(x_j + x_i)). This identity is regarded as a Pfaffian counterpart of Frobenius identity, which is an elliptic generalization of Cauchy's determinant identity for det (1/ (x_i + x_j)).

math.CA↗
arXiv2004

Enumeration of Symmetry Classes of Alternating Sign Matrices and Characters of Classical Groups

An alternating sign matrix is a square matrix with entries 1, 0 and -1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the symmetry classes of alternating sign matrices and their variations, G. Kuperberg associate square ice models with appropriate boundary conditions, and give determinanat and Pfaffian formulae for the partition functions. In this paper, we utilize several determinant and Pfaffian identities to evaluate Kuperberg's determinants and Pfaffians, and express the round partition functions in terms of irreducible characters of classical groups. In particular, we settle a conjecture on the number of vertically and horizontally symmetric alternating sign matrices (VHSASMs).

math.CO↗
arXiv1997

The number of rhombus tilings of a "punctured" hexagon and the minor summation formula

We compute the number of all rhombus tilings of a hexagon with sides $a,b+1,c,a+1,b,c+1$, of which the central triangle is removed, provided $a,b,c$ have the same parity. The result is a product of four numbers, each of which counts the number of plane partitions inside a given box. The proof uses nonintersecting lattice paths and a new identity for Schur functions, which is proved by means of the minor summation formula of Ishikawa and Wakayama. A symmetric generalization of this identity is stated as a conjecture.

math.CO↗