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Arpita Nayek

Publications and source records attributed to Arpita Nayek.

10 recordsLinked to original sources
arXiv2026

Toric Schubert Varieties in Partial Flag Varieties

In this article, we investigate the toric Schubert varieties in partial flag varieties $G/P$ for a connected semisimple algebraic group $G$. Using Deodhar's decomposition of Richardson varieties and the work of Pasquier, we give an explicit description of the fan of a toric Schubert variety, leading to a combinatorial model for its cones. As an application, we obtain necessary and sufficient conditions for smoothness of toric Schubert varieties in terms of the Cartan integers associated to a reduced expression. Furthermore, we prove that for a Coxeter-type element $w \in W^P$, the interval $[e,w]_{W^P}$ is a supersolvable join-distributive lattice. Finally, we apply these results to the study of spherical and horospherical Schubert varieties, providing a combinatorial method for checking the smoothness via the associated toric Schubert varieties.

math.CO↗
arXiv2024

On Automorphism group of a $G$-induced variety

Let $G$ be a connected semisimple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers and $B$ be a Borel subgroup of $G.$ Let $F$ be an irreducible projective $B$-variety. Then consider the variety $E:=G\times^{B}F,$ which has a natural action of $G$; we call it $G$-induced variety or $(G,B)$-induced variety. In this article, we compute the connected component containing the identity automorphism of the group of all algebraic automorphisms of some particular $G$-induced varieties $E.$

math.AG↗
arXiv2024

Torus quotient of the Grassmannian $G_{n,2n}$

Let $G_{n,2n}$ be the Grassmannian parameterizing the $n$-dimensional subspaces of $\mathbb{C}^{2n}.$ The Picard group of $G_{n,2n}$ is generated by a unique ample line bundle $\mathcal{O}(1).$ Let $T$ be a maximal torus of $SL(2n,\mathbb{C})$ which acts on $G_{n,2n}$ and $\mathcal{O}(1).$ By \cite[Theorem 3.10, p.764]{Kum}, $2$ is the minimal integer $k$ such that $\mathcal{O}(k)$ descends to the GIT quotient. In this article, we prove that the GIT quotient of $G_{n,2n}$ ($n\ge 3$) by $T$ with respect to $\mathcal{O}(2)=\mathcal{O}(1)^{\otimes 2}$ is not projectively normal when polarized with the descent of $\mathcal{O}(2).$

math.AG↗
arXiv2023

Torus quotients of Richardson varieties in $G_{r,qr+1}$

Let $r$ and $q$ be positive integers and $n=qr+1.$ Let $G = SL(n, \mathbb{C})$ and $T$ be a maximal torus of $G.$ Let $P^{α_r}$ be the maximal parabolic subgroup of $G$ corresponding to the simple root $α_r.$ Let $ω_r$ be the fundamental weight corresponding to $α_r.$ Let $W$ be the Weyl group of $G$ and $W_{P^{α_r}}$ be the Weyl group of $P^{α_r}.$ Let $W^{P^{α_r}}$ be the set of all minimal coset representatives of $W/W_{P^{α_r}}$ in $W.$ Let $w_{r,n}$ (respectively, $v_{r,n}$) be the minimal (respectively, maximal) element in $W^{P^{α_{r}}}$ such that $w_{r,n}(nω_r) \leq 0$ (respectively, $v_{r,n}(nω_r) \geq 0$). Let $v \leq v_{r,n}$ and $X^v_{w_{r,n}}$ be the Richardson variety in $G_{r,n}$ corresponding to $v$ and $w_{r,n}.$ In this article, we give a sufficient condition on $v$ such that the GIT quotient of $X^{v}_{w_{r,n}}$ for the action of $T$ is the product of projective spaces with respect to the descent of the line bundle $\mathcal{L}(nω_r).$

math.AG↗
arXiv2023

Torus Quotients of Richardson Varieties

For $1\le r\le n-1,$ let $G_{r,n}$ denote the Grassmannian parametrizing $r$-dimensional subspaces of $\mathbb{C}^{n}.$ Let $(r,n)=1.$ In this article we show that the GIT quotients of certain Richardson varieties in $G_{r,n}$ for the action of a maximal torus in $SL(n,\mathbb{C})$ are the product of projective spaces with respect to the descent of a suitable line bundle.

math.AG↗
arXiv2023

On torus quotients of Schubert varieties in orthogonal Grassmannian-II

Let $G=SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the simple root $α_{4n+2}$. In this article, we prove projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle.

math.AG↗
arXiv2022

On torus quotients of Schubert varieties in Orthogonal Grassmannian

Let $G=Spin(8n, \mathbb{C})(n\ge 1)$ and $T_{G}$ be a maximal torus of $G.$ Let $P^{α_{4n}}(\supset T_{G})$ be the maximal parabolic subgroup of $G$ corresponding to the simple root $α_{4n}.$ Let $X$ be a Schubert variety in $G/P^{α_{4n}}$ admitting semi-stable point with respect to the $T$-linearized very ample line bundle $\mathcal{L}(2ω_{4n}).$ Let $R=\bigoplus_{k \in \mathbb{Z}_{\geq 0}}R_k,$ where $R_k=H^{0}(X, \mathcal{L}^{\otimes k}(2ω_{4n}))^{T_{G}}.$ In this article, we prove that for $n=1$ and $X=G/P^{α_4},$ the graded $\mathbb{C}$-algebra $R$ is generated by $R_1.$ As a consequence, we prove that the GIT quotient of $G/P^{α_{4}}$ is projectively normal with respect to the descent of the $T_{G}$-linearized very ample line bundle $\mathcal{L}(2ω_{4})$ and is isomorphic to the projective space $(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(1))$ as a polarized variety. Further, we prove that $R$ is generated by $R_1$ and $R_2$ for some Schubert varieties in $G/P^{α_{4n}}$ (for $n \geq 2$). As a consequence, we prove that the GIT quotient of those Schubert varieties are projectively normal with respect to the descent of the $T_G$-linearized very ample line bundle $\mathcal{L}(4ω_{4n}).$ Moreover, for $G = Spin(2n,\mathbb{C})(n \ge 4)$ (respectively, $G=Sp(2n, \mathbb{C}) (n\ge 2)$) and a maximal torus $T_G$ of $G,$ we prove that the GIT quotient of $G/P^{α_{1}}$ is projectively normal with respect to the descent of the $T_G$-linearized very ample line bundle $\mathcal{L}(2ω_{1})$ and is isomorphic to the projective space $(\mathbb{P}^{n-2},\mathcal{O}_{\mathbb{P}^{n-2}}(1))$ (respectively, $(\mathbb{P}^{n-1},\mathcal{O}_{\mathbb{P}^{n-1}}(1))$ as a polarized variety.

math.AG↗
arXiv2021

Torus quotients of Schubert varieties in the Grassmannian $G_{2,n}$

Let $G=SL(n, \mathbb{C}),$ and $T$ be a maximal torus of $G,$ where $n$ is a positive even integer. In this article, we study the GIT quotients of the Schubert varieties in the Grassmannian $G_{2,n}.$ We prove that the GIT quotients of the Richardson varieties in the minimal dimensional Schubert variety admitting stable points in $G_{2,n}$ are projective spaces. Further, we prove that the GIT quotients of certain Richardson varieties in $G_{2,n}$ are projective toric varieties. Also, we prove that the GIT quotients of the Schubert varieties in $G_{2,n}$ have at most finite set of singular points. Further, we have computed the exact number of singular points of the GIT quotient of $G_{2,n}.$

math.AG↗
arXiv2019

Projective normality of torus quotients of flag varieties

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 $\mathcal{L}_{4\varpi_3}$.

math.RT↗