arXiv subjects

Arkadev Ghosh

Publications and source records attributed to Arkadev Ghosh.

3 recordsLinked to original sources

arXiv2026

Geometric Invariant Theory of Peterson Varieties

We study the GIT quotients of the Peterson variety $\mathrm{Pet}_n\subset \mathrm{GL}(n,\mathbb C)/B$ under a one-parameter subgroup $\lambda:\mathbb G_m \to T$ with respect to the linearization $\mathcal L(\chi)$ given by a regular dominant character $\chi$ in the root lattice. Using the Richardson stratification, we describe the semistable and stable loci explicitly in terms of subsets of simple roots. This determines the GIT chamber decomposition and the corresponding wall-crossing morphisms. In the deep chamber, the quotient is shown to be isomorphic to the weighted projective space $\mathbb P(1,2,\ldots,n-1)$. We obtain a complete chamber-theoretic characterization of normality and describe how normality varies with the choice of linearization. We also prove that the quotient is smooth if and only if $n\le3$, independently of the regular dominant linearization. These results describe how the singular geometry of the Peterson variety is reflected in the variation of its GIT quotients.

math.AG↗

arXiv2026

GIT quotient of minimal dimensional Schubert variety modulo a subtorus

Let $G=PSL(n,\mathbb{C})$. Let $T$ be a maximal torus of $G$. Let $\omega_{r}$ denote the $r^{th}$ fundamental weight. Let $\mathcal{L}(n\omega_{r})$ denote the line bundle on the Grassmannian $G_{r,n}$ associated to the character $n\omega_{r}$ of $T$. In an earlier work of Kannan and Sardar, it is proved that there is a unique minimal dimensional Schubert variety $X(w_{r,n})$ in $G_{r,n}$ admitting semistable points for the $T$-linearized ample line bundle $\mathcal{L}(n\omega_{r})$. Assume that $n=rq+1$, where $r,q\in\mathbb{N}$ and $q\geq 2$. In this paper, we study the GIT quotient of $X(w_{r,n})$ modulo a subtorus $T_{J_{r}}$ of $T$ generated by the one parameter subgroups of $T$ corresponding to the peaks of $w_{r,n}$. We prove that the GIT quotient of $X(w_{r,n})$ modulo $T_{J_{r}}$ is isomorphic to the total space of the $r^{th}$ stage of an iterated projective space bundle over $\mathbb{P}^{q-1}$.

math.AG↗

arXiv2024

GIT quotient of Schubert varieties modulo one dimensional torus

Let $G$ be a simple algebraic group of adjoint type of rank $n$ over $\mathbb{C}$. Let $T$ be a maximal torus of $G$, and $B$ be a Borel subgroup of $G$ containing $T$. Let $W=N_{G}(T)/T$ be the Weyl group of $G$. Let $S=\{\alpha_{1},\ldots,\alpha_{n}\}$ be the set of simple roots of $G$ relative to $(B,T)$. Let $\lambda_{s}$ be the one parameter subgroup of $T$ dual to $\alpha_{s}$. In this paper, we give a criterion for Schubert varieties admitting semistable points for the $\lambda_{s}$-linearized line bundles $\mathcal{L}(\chi)$ associated to every dominant character $\chi$ of $T$. If $\omega_{r}$ is a minuscule fundamental weight and $m\omega_{r}\in X(T)$, then we prove that there is a unique minimal dimensional Schubert variety $X(w_{s,r})$ in $G/P_{S\setminus\{\alpha_{r}\}}$ such that $X(w_{s,r})^{ss}_{\lambda_{s}}(\mathcal{L}(m\omega_{r}))\neq \phi$. Further, we prove that if $G=PSL(n,\mathbb{C})$, and $n\nmid rs$, $m=\frac{n}{(rs,n)}$, and $p=\lfloor\frac{rs}{n}\rfloor$ then the GIT quotient of the minimal dimensional Schubert variety $X(w_{s,r})$ is isomorphic to the projective space $\mathbb{P}(M(s-p, r-p))$, where $M(s-p, r-p)$ is the $(s-p)\times (r-p)$-matrices with complex numbers as entries.

math.AG↗