S S Kannan
Articles written in Proceedings – Mathematical Sciences
Volume 119 Issue 1 February 2009 pp 81-100
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 $GL_{n+1}(k)/B_{n+1}$.
Volume 119 Issue 4 September 2009 pp 469-485
In this paper, for any simple, simply connected algebraic group 𝐺 of type $B,C$ or 𝐷 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 $G/P$. 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 $X(\tau)$ admits a semistable point for the action of the torus 𝑇 with respect to a non-trivial line bundle on $G/B$.
Volume 121 Issue 1 February 2011 pp 19-26
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.
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