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Register a new userThe $(mathfrak{gl}_m,mathfrak{gl}_n$) duality in the quantum toroidal setting
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On a Fock space constructed from $mn$ free bosons and lattice ${Bbb {Z}}^{mn}$, we give a level $n$ action of the quantum toroidal algebra $mathscr {E}_m$ associated to $mathfrak{gl}_m$, together with a level $m$ action of the quantum toroidal algebra ${mathscr E}_n$ associated to ${mathfrak {gl}}_n$. We prove that the $mathscr {E}_m$ transfer matrices commute with the $mathscr {E}_n$ transfer matrices after an appropriate identification of parameters.
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The affine evaluation map is a surjective homomorphism from the quantum toroidal ${mathfrak {gl}}_n$ algebra ${mathcal E}_n(q_1,q_2,q_3)$ to the quantum affine algebra $U_qwidehat{mathfrak {gl}}_n$ at level $kappa$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^n=kappa^2$. We discuss ${mathcal E}_n(q_1,q_2,q_3)$ evaluation modules. We give highest weights of evaluation highest weight modules. We also obtain the decomposition of the evaluation Wakimoto module with respect to a Gelfand-Zeitlin type subalgebra of a completion of ${mathcal E}_n(q_1,q_2,q_3)$, which describes a deformation of the coset theory $widehat{mathfrak {gl}}_n/widehat{mathfrak {gl}}_{n-1}$.
In third paper of the series we construct a large family of representations of the quantum toroidal $gl_1$ algebra whose bases are parameterized by plane partitions with various boundary conditions and restrictions. We study the corresponding formal characters. As an application we obtain a Gelfand-Zetlin type basis for a class of irreducible lowest weight $gl_infty$-modules.
We define a $mathfrak{gl}_N$-stratification of the Grassmannian of $N$ planes $mathrm{Gr}(N,d)$. The $mathfrak{gl}_N$-stratification consists of strata $Omega_{mathbf{Lambda}}$ labeled by unordered sets $mathbf{Lambda}=(lambda^{(1)},dots,lambda^{(n)})$ of nonzero partitions with at most $N$ parts, satisfying a condition depending on $d$, and such that $(otimes_{i=1}^n V_{lambda^{(i)}})^{mathfrak{sl}_N} e 0$. Here $V_{lambda^{(i)}}$ is the irreducible $mathfrak{gl}_N$-module with highest weight $lambda^{(i)}$. We show that the closure of a stratum $Omega_{mathbf{Lambda}}$ is the union of the strata $Omega_{mathbfXi}$, $mathbf{Xi}=(xi^{(1)},dots,xi^{(m)})$, such that there is a partition ${I_1,dots,I_m}$ of ${1,2,dots,n}$ with $ {rm {Hom}}_{mathfrak{gl}_N} (V_{xi^{(i)}}, otimes_{jin I_i}V_{lambda^{(j)}}big) eq 0$ for $i=1,dots,m$. The $mathfrak{gl}_N$-stratification of the Grassmannian agrees with the Wronski map. We introduce and study the new object: the self-dual Grassmannian $mathrm{sGr}(N,d)subset mathrm{Gr}(N,d)$. Our main result is a similar $mathfrak{g}_N$-stratification of the self-dual Grassmannian governed by representation theory of the Lie algebra $mathfrak {g}_{2r+1}:=mathfrak{sp}_{2r}$ if $N=2r+1$ and of the Lie algebra $mathfrak g_{2r}:=mathfrak{so}_{2r+1}$ if $N=2r$.
In 1990 Beilinson, Lusztig and MacPherson provided a geometric realization of modified quantum $mathfrak{gl}_n$ and its canonical basis. A key step of their work is a construction of a monomial basis. Recently, Du and Fu provided an algebraic construction of the canonical basis for modified quantum affine $mathfrak{gl}_n$, which among other results used an earlier construction of monomial bases using Ringel-Hall algebra of the cyclic quiver. In this paper, we give an elementary algebraic construction of a monomial basis for affine Schur algebras and modified quantum affine $mathfrak{gl}_n$.
We begin a study of the representation theory of quantum continuous $mathfrak{gl}_infty$, which we denote by $mathcal E$. This algebra depends on two parameters and is a deformed version of the enveloping algebra of the Lie algebra of difference operators acting on the space of Laurent polynomials in one variable. Fundamental representations of $mathcal E$ are labeled by a continuous parameter $uin {mathbb C}$. The representation theory of $mathcal E$ has many properties familiar from the representation theory of $mathfrak{gl}_infty$: vector representations, Fock modules, semi-infinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from $mathcal E$ to spherical double affine Hecke algebras $Sddot H_N$ for all $N$. A key step in this construction is an identification of a natural bases of the tensor products of vector representations with Macdonald polynomials. We also show that one of the Fock representations is isomorphic to the module constructed earlier by means of the $K$-theory of Hilbert schemes.
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