arXiv subjects
Publications and source records attributed to Anatol N. Kirillov.
We construct a hook-content formula and its $q$-analog using excited Young diagrams analogous to Naruse's hook-length formula for skew shapes. Furthermore, we show that our hook-content formula has a simple factorization and give some conjectures and questions related to its $q$-analog.
For any given loopless graph $G$, we introduce $Q$ - deformations of its Postnikov-Shapiro algebras counting spanning trees, counting spanning forests and $Q$ - deformations of internal algebra of $G$. We determine the total dimension of the algebras; our proof also gives a new proof of the formula for the total dimensions of the usual Postnikov-Shapiro algebras. Furthermore, we construct "square-free" definition of usual internal algebra of $G$.
We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.
We propose a method to determine the quantum numbers, which we call the rigged configurations, for the solutions to the Bethe ansatz equations for the spin-1/2 isotropic Heisenberg model under the periodic boundary condition. Our method is based on the observation that the sums of Bethe's quantum numbers within each string behave particularly nicely. We confirm our procedure for all solutions for length 12 chain (totally 923 solutions).
We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations. One can find more details about the content of present paper in Extended Abstract.
We compute the energy eigenvalues of Nepomechie--Wang's eigenstates for the spin 1/2 isotropic Heisenberg chain.
We provide a conjecture for the following two quantities related with the spin-$\frac{1}{2}$ isotropic Heisenberg model defined over rings of even lengths: (i) the number of the solutions to the Bethe ansatz equations which correspond to non-zero Bethe vectors; (ii) the number of physical singular solutions of the Bethe ansatz equations in the sense of Nepomechie-Wang. The conjecture is based on a natural relationship between the solutions to the Bethe ansatz equations and the rigged configurations.
The generating function for elements of the Bethe subalgebra of Hecke algebra is constructed as Sklyanin's transfer-matrix operator for Hecke chain. We show that in a special classical limit q -> 1 the Hamiltonians of the Gaudin model can be derived from the transfer-matrix operator of Hecke chain. We consruct a non-local analogue of the Gaudin Hamiltonians for the case of Hecke algebras.
We construct a model of the affine nil-Hecke algebra as a subalgebra of the Nichols-Woronowicz algebra associated to a Yetter-Drinfeld module over the affine Weyl group. We also discuss the Peterson isomorphism between the homology of the affine Grassmannian and the small quantum cohomology ring of the flag variety in terms of the braided differential calculus.
We study quantum entanglement in the ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model defined on two-dimensional graphs with reflection and/or inversion symmetry. The ground state of this spin model is known as the valence-bond-solid state. We investigate the properties of reduced density matrix of a subsystem which is a mirror image of the other one. Thanks to the reflection symmetry, the eigenvalues of the reduced density matrix can be obtained by numerically diagonalizing a real symmetric matrix whose elements are calculated by Monte Carlo integration. We calculate the von Neumann entropy of the reduced density matrix. The obtained results indicate that there is some deviation from the naive expectation that the von Neumann entropy per valence bond on the boundary between the subsystems is $\ln 2$. This deviation is interpreted in terms of the hidden spin chain along the boundary between the subsystems. In some cases where graphs are on ladders, the numerical results are analytically or algebraically confirmed.
We give an interpretation of the t=1 specialization of the modified Macdonald polynomial as a generating function of the energy statistics defined on the set of paths arising in the context of Box-Ball Systems (BBS-paths for short). We also introduce one parameter generalizations of the energy statistics on the set of BBS-paths which all, conjecturally, have the same distribution.
We give several equivalent combinatorial descriptions of the space of states for the box-ball systems, and connect certain partition functions for these models with the q-weight multiplicities of the tensor product of the fundamental representations of the Lie algebra gl(n). As an application, we give an elementary proof of the special case t=1 of the Haglund--Haiman--Loehr formula. Also, we propose a new class of combinatorial statistics that naturally generalize the so-called energy statistics.
A generalized model of Heisenberg quantum antiferromagnet on an arbitrary graph is constructed so that the VBS is the unique ground state. The norm of the base state and equal time multi point correlation functions are computed in terms of generalized hyper geometric functions. For the one-dimensional periodic Heisenberg model we present a method of computing multi point correlation functions based on the study of a commuting family of transfer matrices. The connection of multi point correlators with Young tableaux and Gegenbauer polynomials is found.
There are two distinct approaches to the study of initial value problem of the periodic box-ball systems. One way is the rigged configuration approach due to Kuniba--Takagi--Takenouchi and another way is the 10-elimination approach due to Mada--Idzumi--Tokihiro. In this paper, we describe precisely interrelations between these two approaches.
For the root system of type $A$ we introduce and study a certain extension of the quadratic algebra invented by S. Fomin and the first author, to construct a model for the equivariant cohomology ring of the corresponding flag variety. As an application of our construction we describe a generalization of the equivariant Pieri rule for double Schubert polynomials. For a general finite Coxeter system we construct an extension of the corresponding Nichols-Woronowicz algebra. In the case of finite crystallographic Coxeter systems we present a construction of extended Nichols-Woronowicz algebra model for the equivariant cohomology of the corresponding flag variety.
We discuss a class of generalized divided difference operators which give rise to a representation of Nichols-Woronowicz algebras associated to Weyl groups. For the root system of type $A,$ we also study the condition for the deformations of the Fomin-Kirillov quadratic algebra, which is a quadratic lift of the Nichols-Woronowicz algebra, to admit a representation given by generalized divided difference operators. The relations satisfied by the mutually commuting elements called Dunkl elements in the deformed Fomin-Kirillov algebra are determined. The Dunkl elements correspond to the truncated elliptic Dunkl operators via the representation given by the generalized divided difference operators.
We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We also prove that, under certain assumptions, the skew divided difference operators transform the Schubert polynomials into polynomials with positive integer coefficients.
We give a model of the coinvariant algebra of the complex reflection groups as a subalgebra of a braided Hopf algebra called Nichols-Woronowicz algebra.