arXiv subjects
Publications and source records attributed to Santosha Kumar Pattanayak.
In this paper we study Weyl modules for a toroidal Lie algebra $\CT$ with arbitrary $n$ variables. Using the work of Rao \cite{1995}, we prove that the level one global Weyl modules of $\CT$ are isomorphic to suitable submodules of a Fock space representation of $\CT$ upto a twist. As an application, we compute the graded character of the level one local Weyl module of $\CT$, thereby generalising the work of Kodera \cite{ko}.
In this paper, we consider the GIT quotients of Schubert varieties for the action of a maximal torus. We describe the minuscule Schubert varieties for which the semistable locus is contained in the smooth locus. As a consequence, we study the smoothness of torus quotients of Schubert varieties in the Grassmannian. We also prove that the torus quotient of any Schubert variety in the homogeneous space $SL(n, \mathbb C)/P$ is projectively normal with respect to the line bundle $\mathcal L_{α_0}$ and the quotient space is a projective space, where the line bundle $\mathcal L_{α_0}$ and the parabolic subgroup $P$ of $SL (n, \mathbb C)$ are associated to the highest root $α_0$.
Let $G=SL_n(\mathbb C)$ and $T$ be a maximal torus in $G$. We show that the quotient $T \backslash \backslash G/{P_{α_1}\cap P_{α_2}}$ is projectively normal with respect to the descent of a suitable line bundle, where $P_{α_i}$ is the maximal parabolic subgroup in $G$ associated to the simple root $α_i$, $i=1,2$. We give a degree bound of the generators of the homogeneous coordinate ring of $T \backslash \backslash (G_{3,6})^{ss}_T(\mathcal{L}_{2\varpi_3})$. If $G =Spin_7$, we give a degree bound of the generators of the homogeneous coordinate ring of $T \backslash \backslash (G/P_{α_2})^{ss}_T(\mathcal{L}_{2\varpi_2})$ whereas we prove that the quotient $T\backslash\backslash (G/P_{α_3})^{ss}_T(\mathcal{L}_{4\varpi_3})$ is projectively normal with respect to the descent of the line bundles $\mathcal{L}_{4\varpi_3}$.
For any simple, simply connected algebraic group $G$ of type $B,C$ and $D$ and for any maximal parabolic subgroup $P$ of $G$, we provide a criterion for a Richardson variety in $G/P$ to admit semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$.
In this article, we prove that for any indecomposable dominant character of a maximal torus $T$ of a simple adjoint group $G$ such that there is a Coxeter element $w \in W$ for which $X(w)^{ss}_T(\mathcal L_χ) \neq \emptyset$. If further, for any dominant character $χ_1$ of $T$ such that $χ_1\lneqq χ$ with respect to the dominant ordering, $dim(H^0(G/B, \mathcal L_{χ_1})^T) < dim (H^0(G/B, \mathcal L_χ)^T)$, then the graded algebra $\oplus_{d \in \mathbb Z_{\geq 0}}H^0(G/B, \mathcal L_χ^{\otimes d})^T$ is a polynomial ring in $r$ variables where $r\geq 2$.