We study the explicit formula of Lusztig's integral forms of the level one quantum affine algebra $U_q(\hat{sl}_2)$ in the endomorphism ring of symmetric functions in infinitely many variables tensored with the group algebra of $\mathbb Z$. Schur functions are realized as certain orthonormal basis vectors in the vertex representation associated to the standard Heisenberg algebra. In this picture the Littlewood-Richardson rule is expressed by integral formulas, and is used to define the action of Lusztig's $\mathbb Z[q, q]$-form of $U_q(\hat{sl}_2)$ on Schur polynomials.
Affine analogue of Jack's polynomials introduced by Etingof and Kirillov Jr. is studied for the case of \hat{sl}_2. Using the Wakimoto representation, we give an integral formula of elliptic Selberg type for the affine Jack's polynomials. From this integral formula, the action of SL_2(Z) on the space of the affine Jack's polynomials is computed. For simple cases, we write down the affine Jack's polynomials in terms of some modular and elliptic functions.
The decompositions of the Fock and Basic modules of the affine Lie algebra \hat{sl(N)} into irreducible submodules of the Yangian algebra Y(gl(N)) are constructed. Each of the irreducible submodules admits the unique up to normalization eigenbasis of the maximal commutative subalgebra of the Yangian. The elements of this eigenbasis are identified with specializations of Macdonald symmetric functions where both parameters of these functions approach an N-th primitive root of unity.
We consider the double affine Hecke algebra $H=H(k_0,k_1,k^\vee_0,k^\vee_1;q)$ associated with the root system $(C^\vee_1,C_1)$. We display three elements $x$, $y$, $z$ in $H$ that satisfy essentially the $Z_3$-symmetric Askey-Wilson relations. We obtain the relations as follows. We work with an algebra $\hat H$ that is more general than $H$, called the universal double affine Hecke algebra of type $(C_1^\vee,C_1)$. An advantage of $\hat H$ over $H$ is that it is parameter free and has a larger automorphism group. We give a surjective algebra homomorphism ${\hat H} \to H$. We define some eleme…
An polynomial identity is derived from the representation V_m(x)\otimes V_n(y) of U_q(\hat {sl_2}) and a new basis of V_m(x)\otimes V_n(y) is established under some condition.
The infinite affine Lie algebras of type ABCD, also called $\widehat{\mathfrak{gl}}(\infty)$, $\widehat{\mathfrak{o}}(\infty)$, $\widehat{\mathfrak{sp}}(\infty)$, are equivalent to subalgebras of the quantum $W_{1+\infty}$ algebras. They have well-known representations on the Fock space of either a Dirac fermion ($\hat A_\infty$), a Majorana fermion ($\hat B_\infty$ and $\hat D_\infty$) or a symplectic boson ($\hat C_\infty$). Explicit formulas for the action of the quantum $W_{1+\infty}$ subalgebras on the Fock states are proposed for each representation. These formulas are the equivalent of …
We consider $\hat{sl_2}$ spaces of coinvariants with respect to two kinds of ideals of the enveloping algebra $U(sl_2\otimes\C[t])$. The first one is generated by $sl_2\otimes t^N$, and the second one is generated by $e\otimes P(t), f\otimes R(t)$ where $P(t), R(t)$ are fixed generic polynomials. (We also treat a generalization of the latter.) Using a method developed in our previous paper, we give new fermionic formulas for their Hilbert polynomials in terms of the level-restricted Kostka polynomials and $q$-multinomial symbols. As a byproduct, we obtain a fermionic formula for the fusion pro…
q-Supernomial coefficients are generalizations of the q-binomial coefficients. They can be defined as the coefficients of the Hall-Littlewood symmetric function in a product of the complete symmetric functions or the elementary symmetric functions. Hatayama et al. give explicit expressions for these q-supernomial coefficients. A combinatorial expression as the generating function of ribbon tableaux with (co)spin statistic follows from the work of Lascoux, Leclerc and Thibon. In this paper we interpret the formulas by Hatayama et al. in terms of rigged configurations and provide an explicit sta…
An algebra homomorphism $ψ$ from the nonstandard q-deformed (cyclically symmetric) algebra $U_q(so_3)$ to the extension ${\hat U}_q(sl_2)$ of the Hopf algebra $U_q(sl_2)$ is constructed. Not all irreducible representations of $U_q(sl_2)$ can be extended to representations of ${\hat U}_q(sl_2)$. Composing the homomorphism $ψ$ with irreducible representations of ${\hat U}_q(sl_2)$ we obtain representations of $U_q(so_3)$. Not all of these representations of $U_q(so_3)$ are irreducible. Reducible representations of $U_q(so_3)$ are decomposed into irreducible components. In this way we obtain all …
We give a combinatorial formula for the non-symmetric Macdonald polynomials E_μ(x;q,t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J_μ(x;q,t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop, that characterizes the non-symmetric Macdonald polynomials.
A characterization of the space of symmetric Laurent polynomials of type $(BC)_n$ which vanish on a certain set of submanifolds is given by using the Koornwinder-Macdonald polynomials. A similar characterization was given previously for symmetric polynomials of type $A_n$ by using the Macdonald polynomials. We use a new method which exploits the duality relation. The method simplifies a part of the proof in the $A_n$ case.
The X=M conjecture of Hatayama et al. asserts the equality between the one-dimensional configuration sum X expressed as the generating function of crystal paths with energy statistics and the fermionic formula M for all affine Kac--Moody algebra. In this paper we prove the X=M conjecture for tensor products of Kirillov--Reshetikhin crystals B^{1,s} associated to symmetric powers for all nonexceptional affine algebras.
On the level-1 Fock space modules of the algebra $U_q(\hat{sl_n})$ we define a level-0 action $U_0$ of the $U_q(\hat{sl_n})$, and an action of an abelian algebra of conserved Hamiltonians commuting with the $U_0$. An irreducible decomposition of the Fock space with respect to the level-0 action is derived by constructing a base of the Fock space in terms of the Non-symmetric Macdonald Polynomials.
Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly in a suitable semi-classical limit, as $\hbar \to 0$, and becomes non-commutative or "quantum" away from this limit. For a classical curve defined by the zero locus of a polynomial $A(x,y)$, we provide a construction of its non-commutative counterpart $\hat{A} (\hat x, \hat y)$ using the technique of the topological recursion. This leads to a powerful and sys…
Let $\hat{A}_n$ be the completion by the degree of a differential operator of the $n$-th Weyl algebra with generators $x_1,\ldots,x_n,\partial^1,\ldots,\partial^n$. Consider $n$ elements $X_1,\ldots,X_n$ in $\hat{A}_n$ of the form $$ X_i = x_i + \sum_{K = 1}^\infty \sum_{l = 1}^n\sum_{j = 1}^n x_l p_{ij}^{K-1,l}(\partial)\partial^j, $$ where $p^{K-1,l}_{ij}(\partial)$ is a degree $(K-1)$ homogeneous polynomial in $\partial^1,\ldots,\partial^n$, antisymmetric in subscripts $i,j$. Then for any natural $k$ and any function $i : \{1,\ldots,k\}\to\{1,\ldots,n\}$ we prove $$ \sum_{σ\in Σ(k)} X_{i_{σ…
Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. By definition a Leonard pair on $V$ is a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy the following two conditions: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. There is a correspondence between Leonard pa…
The symmetric Macdonald polynomials are able to be constructed out of the non-symmetric Macdonald polynomials. This allows us to develop the theory of the symmetric Macdonald polynomials by first developing the theory of their non-symmetric counterparts. In taking this approach we are able to obtain new results as well as simpler and more accessible derivations of a number of the known fundamental properties of both kinds of polynomials.
We study a class of representations of the Lie algebra of Laurent polynomials with values in the nilpotent subalgebra of sl(3). We derive Weyl-type (bosonic) character formulas for these representations. We establish a connection between the bosonic formulas and the Whittaker vector in the Verma module for the quantum group $U_v sl(3)$. We also obtain a fermionic formula for an eigenfunction of the sl(3) quantum Toda Hamiltonian.
The theory of non-symmetric Jack polynomials is developed independently of the theory of symmetric Jack polynomials, and this theory together with the relationship between the non-symmetric, symmetric and anti-symmetric Jack polynomials is used to deduce the corresponding results for the symmetric Jack polynomials.
We provide elementary identities relating the three known types of non-symmetric interpolation Macdonald polynomials. In addition we derive a duality for non-symmetric interpolation Macdonald polynomials. We consider some applications of these results, in particular for binomial formulas involving non-symmetric interpolation Macdonald polynomials.
The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial $A(x,y)$. Another "family version" of the volume conjecture depends on a quantization parameter, usually denoted $q$ or $\hbar$; this quantum volume conjecture (also known as the AJ-conjecture) can be stated in a form of a q-difference equation that annihilates the colored Jones polynomials and $SL(2,\C)$ Chern-Simons partition functions. We propose refinements / categorifications of both conjectures that include a…
We propose a new realization of the elliptic quantum group equipped with the H-Hopf algebroid structure on the basis of the elliptic algebra U_{q,p}(\hat{sl}_2). The algebra U_{q,p}(\hat{sl}_2) has a constructive definition in terms of the Drinfeld generators of the quantum affine algebra U_q(\hat{sl}_2) and a Heisenberg algebra. This yields a systematic construction of both finite and infinite-dimensional dynamical representations and their parallel structures to U_q(\hat{sl}_2). In particular we give a classification theorem of the finite-dimensional irreducible pseudo-highest weight represe…
Let B_{(l)} be the perfect crystal for the l-symmetric tensor representation of the quantum affine algebra U'_q(\hat{sl(n)}). For a partition mu = (mu_1,...,mu_m), elements of the tensor product B_{(mu_1)} \otimes ... \otimes B_{(mu_m)} can be regarded as inhomogeneous paths. We establish a bijection between a certain large mu limit of this crystal and the crystal of an (generally reducible) integrable U_q(\hat{sl(n)})-module, which forms a large family depending on the inhomogeneity of mu kept in the limit. For the associated one dimensional sums, relations with the Kostka-Foulkes polynomials…
We extend the previous paper "Macdonald's evaluation ... and applications" to the non-symmetric polynomilas recently introduced by Macdonald (as difference counterparts of Opdam's non-symmetric ones).
In this paper we give two realizations of the restricted Kostka polynomials for $\sl_2$. Firstly we identify the restricted Kostka polynomials with a characters of the zero homology of the current algebra with a coefficients in a certain modules. As a corollary we reobtain the alternating sum formula. Secondly we show that the restricted Kostka polynomials are a $q$-multiplicities of the decomposition of the certain integrable $\hat{\sl}_2$-modules to the irreducible components. This allows to write a kind of fermionic formula for the Virasoro unitary models.
The physical 3d $\mathcal{N}=2$ theory T[Y] was previously used to predict the existence of some 3-manifold invariants $\hat{Z}_{a}(q)$ that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the roots of unity should recover the Witten-Reshetikhin-Turaev invariants. In this paper we discuss how, for complements of knots in $S^3$, the analogue of the invariants $\hat{Z}_{a}(q)$ should be a two-variable series $F_K(x,q)$ obtained by parametric resurgence from the asymptotic expansion of the colored Jones polynomial. The terms in this ser…
