Self-dual Grassmannian, Wronski map, and representations of $mathfrak{gl}_N$, ${mathfrak{sp}}_{2r}$, ${mathfrak{so}}_{2r+1}$

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 $N=2r$.

On the Gaudin model associated to Lie algebras of classical types

We derive explicit formulas for solutions of the Bethe Ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple.

Bethe Algebra of Homogeneous XXX Heisenberg Model Has Simple Spectrum

We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional $gl_2$-modules. As a byproduct we show that there exist exactly $binom {n}{l}-binom{n}{l-1}$ two-dimensional vector subspaces $V subset C[u]$ with a basis $f,gin V$ such that $deg f = l, deg g = n-l+1$ and $f(u)g(u-1) - f(u-1)g(u) = (u+1)^n$.

The gl_2 Bethe algebra associated with a nilpotent element

To any 2x2-matrix K one assigns a commutative subalgebra B^{K}subset U(gl_2[t]) called a Bethe algebra. We describe relations between the Bethe algebras, associated with the zero matrix and a nilpotent matrix.

On separation of variables and completeness of the Bethe ansatz for quantum gl_N Gaudin model

In this note, we discuss implications of the results obtained in [MTV4]. It was shown there that eigenvectors of the Bethe algebra of the quantum gl_N Gaudin model are in a one-to-one correspondence with Fuchsian differential operators with polynomial kernel. Here, we interpret this fact as a separation of variables in the gl_N Gaudin model. Having a Fuchsian differential operator with polynomial kernel, we construct the corresponding eigenvector of the Bethe algebra. It was shown in [MTV4] that the Bethe algebra has simple spectrum if the evaluation parameters of the Gaudin model are generic. In that case, our Bethe ansatz construction produces an eigenbasis of the Bethe algebra.