An (inverse) Pieri formula for Macdonald polynomials of type C

We give an explicit Pieri formula for Macdonald polynomials attached to the root system C_n (with equal multiplicities). By inversion we obtain an explicit expansion for two-row Macdonald polynomials of type C.Comment: 31 pages, LaTeX, to appear in Transformation Group

Deligne categories and reduced Kronecker coefficients

The Kronecker coefficients are the structural constants for the tensor categories of representations of the symmetric groups, namely, given three partitions λ,μ,τ of n, the multiplicity of λ in μ⊗τ is called the Kronecker coefficient g[superscript λ][subscript μ,τ]. When the first part of each of the partitions is taken to be very large (the remaining parts being fixed), the values of the appropriate Kronecker coefficients stabilize; the stable value is called the reduced (or stable) Kronecker coefficient. These coefficients also generalize the Littlewood–Richardson coefficients and have been studied quite extensively. In this paper, we show that reduced Kronecker coefficients appear naturally as structure constants of Deligne categories [bar under Rep](S[subscript t]). This allows us to interpret various properties of the reduced Kronecker coefficients as categorical properties of Deligne categories [bar under Rep](S[subscript t]) and derive new combinatorial identities

A Selberg integral for the Lie algebra A_n

A new q-binomial theorem for Macdonald polynomials is employed to prove an A_n analogue of the celebrated Selberg integral. This confirms the g=A_n case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra g.Comment: 32 page

Tests of Seiberg-like Duality in Three Dimensions

We use localization techniques to study several duality proposals for supersymmetric gauge theories in three dimensions reminiscent of Seiberg duality. We compare the partition functions of dual theories deformed by real mass terms and FI parameters. We find that Seiberg-like duality for N=3 Chern-Simons gauge theories proposed by Giveon and Kutasov holds on the level of partition functions and is closely related to level-rank duality in pure Chern-Simons theory. We also clarify the relationship between the Giveon-Kutasov duality and a duality in theories of fractional M2 branes and propose a generalization of the latter. Our analysis also confirms previously known results concerning decoupled free sectors in N=4 gauge theories realized by monopole operators.Comment: 36 pages, 5 figure

Semi-invariants of symmetric quivers of finite type

Let $(Q,\sigma)$ be a symmetric quiver, where $Q=(Q_0,Q_1)$ is a finite quiver without oriented cycles and $\sigma$ is a contravariant involution on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form on a representation $V$ of $Q$. We shall call the representation orthogonal if is symmetric and symplectic if $$ is skew-symmetric. Moreover we can define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. For symmetric quivers of finite type, we prove that the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type $c^V$ and, in the case when matrix defining $c^V$ is skew-symmetric, by the Pfaffians $pf^V$

Semi-invariants of symmetric quivers of tame type

A symmetric quiver $(Q,\sigma)$ is a finite quiver without oriented cycles $Q=(Q_0,Q_1)$ equipped with a contravariant involution $\sigma$ on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form on a representation $V$ of $Q$. We shall say that $V$ is orthogonal if is symmetric and symplectic if $$ is skew-symmetric. Moreover, we define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. So we prove that if $(Q,\sigma)$ is a symmetric quiver of tame type then the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type $c^V$ and, when matrix defining $c^V$ is skew-symmetric, by the Pfaffians $pf^V$. To prove it, moreover, we describe the symplectic and orthogonal generic decomposition of a symmetric dimension vector

Schur Q-functions and degeneracy locus formulas for morphisms with symmetries

We give closed-form formulas for the fundamental classes of degeneracy loci associated with vector bundle maps given locally by (not necessary square) matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal. Our description uses essentially Schur Q-polynomials of a bundle, and is based on a certain push-forward formula for these polynomials in a Grassmann bundle.Comment: 22 pages, AMSTEX, misprints corrected, exposition improved. to appear in the Proceedings of Intersection Theory Conference in Bologna, "Progress in Mathematics", Birkhause

Correlation Functions in 2-Dimensional Integrable Quantum Field Theories

In this talk I discuss the form factor approach used to compute correlation functions of integrable models in two dimensions. The Sinh-Gordon model is our basic example. Using Watson's and the recursive equations satisfied by matrix elements of local operators, I present the computation of the form factors of the elementary field $\phi(x)$ and the stress-energy tensor $T_{\mu\nu}(x)$ of the theory.Comment: 19pp, LATEX version, (talk at Como Conference on ``Integrable Quantum Field Theories''

Partial domain wall partition functions

We consider six-vertex model configurations on an n-by-N lattice, n =< N, that satisfy a variation on domain wall boundary conditions that we define and call "partial domain wall boundary conditions". We obtain two expressions for the corresponding "partial domain wall partition function", as an (N-by-N)-determinant and as an (n-by-n)-determinant. The latter was first obtained by I Kostov. We show that the two determinants are equal, as expected from the fact that they are partition functions of the same object, that each is a discrete KP tau-function, and, recalling that these determinants represent tree-level structure constants in N=4 SYM, we show that introducing 1-loop corrections, as proposed by N Gromov and P Vieira, preserves the determinant structure.Comment: 30 pages, LaTeX. This version, which appeared in JHEP, has an abbreviated abstract and some minor stylistic change