arXiv subjects

S. S. Kannan

Publications and source records attributed to S. S. Kannan.

8 recordsLinked to original sources

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 $ω_{r}$ denote the $r^{th}$ fundamental weight. Let $\mathcal{L}(nω_{r})$ denote the line bundle on the Grassmannian $G_{r,n}$ associated to the character $nω_{r}$ 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 $\mathcal{L}(nω_{r})$. 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 $S=\{α_{1},\ldots,α_{n}\}$ be the set of simple roots of $G$ relative to $(B,T)$. Let $λ_{s}$ be the one parameter subgroup of $T$ dual to $α_{s}$. In this paper, we give a criterion for Schubert varieties admitting semistable points for the $λ_{s}$-linearized line bundles $\mathcal{L}(χ)$ associated to every dominant character $χ$ of $T$. If $ω_{r}$ is a minuscule fundamental weight and $mω_{r}\in X(T)$, then we prove that there is a unique minimal dimensional Schubert variety $X(w_{s,r})$ in $G/P_{S\setminus\{α_{r}\}}$ such that $X(w_{s,r})^{ss}_{λ_{s}}(\mathcal{L}(mω_{r}))\neq ϕ$. 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↗

arXiv2010

Projective normality of Weyl group quotients

In this note, we prove that for the standard representation $V$of the Weyl group $W$ of a semi-simple algebraic group of type $A_n, B_n, C_n, D_n, F_4$ and $G_2$ over $\mathbb C$, the projective variety $\mathbb P(V^m)/W$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |W|}$, where $V^m$ denote the direct sum of $m$ copies of $V$. We also prove that for any finite group $G$ and for any finite dimentional representation $V$ over $\mathbb C$, the projective variety $P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes n!}$ as a consequence.

math.AG↗

arXiv2009

Projective normality of finite group quotients and EGZ theorem

In this note, we prove that for any finite dimensional vector space $V$ over $\mathbb {C}$, and for a finite cyclic group $G$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |G|}$ by a method using toric variety, and deduce the EGZ theorem as a consequence.

math.AG↗

arXiv2009

Nondegeneracy for Quotient Varieties under Finite Group Actions

We prove that for an abelian group $G$ of order $n$ the morphism $ φ\colon \mathbf{P}(V^*)\longrightarrow \mathbf{P} ((\mathrm{sym}^n V^*)^G)$ defined by $φ([f]) = [\prod_{σ\in G} σ\cdot f ]$ is nondegenerate for every finite-dimensional representation $V$ of $G$ if and only if either $n$ is a prime number or $n=4$.

math.AG↗

arXiv2008

Torus quotients of homogeneous spaces-minimal dimensional Schubert Variety admitting semi-stable points

In this paper, for any simple, simply connected algebraic group $G$ of type $B_n,C_n$ or $D_n$ and for any maximal parabolic subgroup $P$ of $G$, we describe all minimal dimensional Schubert varieties in $G/P$ admitting semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$. In this paper, we also describe, for any semi-simple simply connected algebraic group $G$ and for any Borel subgroup $B$ of $G$, all Coxeter elements $τ$ for which the Schubert variety $X(τ)$ admits a semistable point for the action of the torus $T$ with respect to a non-trivial line bundle on $G/B$.

math.RT↗

arXiv2008

Projective normality of quotient varieties modulo finite groups

In this note, we prove that for any finite dimensional vector space $V$ over an algebraically closed field $k$, and for any finite subgroup $G$ of $GL(V)$ which is either solvable or is generated by pseudo reflections such that the $|G|$ is a unit in $k$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |G|}$.

math.AG↗

arXiv2007

Torus quotients of homogeneous spaces of the general linear group and the standard representation of certain symmetric groups

We give a stratification of the GIT quotient of the Grassmannian $G_{2,n}$ modulo the normaliser of a maximal torus of $SL_{n}(k)$ with respect to the ample generator of the Picard group of $G_{2,n}$. We also prove that the flag variety $GL_{n}(k)/B_{n}$ can be obtained as a GIT quotient of $GL_{n+1}(k)/B_{n+1}$ modulo a maximal torus of $SL_{n+1}(k)$ for a suitable choice of an ample line bundle on $GL_{n+1}(k)/B_{n+1}$.

math.AG↗