Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSelf-dual Grassmannian, Wronski map, and representations of $mathfrak{gl}_N$, ${mathfrak{sp}}_{2r}$, ${mathfrak{so}}_{2r+1}$
146
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $N=2r$.
rate research
Read More
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.
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 $widehat{mathfrak {gl}}_n/widehat{mathfrak {gl}}_{n-1}$.
We show that the Kazhdan-Lusztig category $KL_k$ of level-$k$ finite-length modules with highest-weight composition factors for the affine Lie superalgebra $widehat{mathfrak{gl}(1|1)}$ has vertex algebraic braided tensor supercategory structure, and that its full subcategory $mathcal{O}_k^{fin}$ of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple $widehat{mathfrak{gl}(1|1)}$-module in $KL_k$ has a projective cover in $mathcal{O}_k^{fin}$, and we determine all fusion rules involving simple and projective objects in $mathcal{O}_k^{fin}$. Then using Knizhnik-Zamolodchikov equations, we prove that $KL_k$ and $mathcal{O}_k^{fin}$ are rigid. As an application of the tensor supercategory structure on $mathcal{O}_k^{fin}$, we study certain module categories for the affine Lie superalgebra $widehat{mathfrak{sl}(2|1)}$ at levels $1$ and $-frac{1}{2}$. In particular, we obtain a tensor category of $widehat{mathfrak{sl}(2|1)}$-modules at level $-frac{1}{2}$ that includes relaxed highest-weight modules and their images under spectral flow.
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.
Let $mathfrak g = mathfrak{gl}_N(k)$, where $k$ is an algebraically closed field of characteristic $p > 0$, and $N in mathbb Z_{ge 1}$. Let $chi in mathfrak g^*$ and denote by $U_chi(mathfrak g)$ the corresponding reduced enveloping algebra. The Kac--Weisfeiler conjecture, which was proved by Premet, asserts that any finite dimensional $U_chi(mathfrak g)$-module has dimension divisible by $p^{d_chi}$, where $d_chi$ is half the dimension of the coadjoint orbit of $chi$. Our main theorem gives a classification of $U_chi(mathfrak g)$-modules of dimension $p^{d_chi}$. As a consequence, we deduce that they are all parabolically induced from a 1-dimensional module for $U_0(mathfrak h)$ for a certain Levi subalgebra $mathfrak h$ of $mathfrak g$; we view this as a modular analogue of M{oe}glins theorem on completely primitive ideals in $U(mathfrak{gl}_N(mathbb C))$. To obtain these results, we reduce to the case $chi$ is nilpotent, and then classify the 1-dimensional modules for the corresponding restricted $W$-algebra.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
