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

Abstract

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 .Let k be an algebraically closed field. Consider the action of a maximal torus T of SL n (k) on the Grassmannian G r,n of r-dimensional vector subspaces of an n-dimensional vector space over k. Let N denote the normaliser of T in SL n (k). Let L r denote the ample generator of the Picard group of G r,n . Let W = N/T denote the Weyl group of SL n (k) with respect to T .In [7], it is shown that the semi-stable points of G r,n with respect to the T -linearised ample line bundle L r is same as the stable points if and only if r and n are co-prime.In this paper, we describe all the semi-stable points of G r,n with respect to L r . In this connection, we prove the following result: First, we introduce some notation needed for the statement of the theorem.Let h j be a Cartan subalgebra of sl j +1 , P(h j ) be the projective space and R j ⊆ h * j be the root system. Let V j be the open subset of P(h j ) defined byHere, the Weyl group of sl j +1 is S j +1 , and h j is the standard representation of S j +1 .With this notation, taking m = n−1 2 (for this notation, see Lemma 1.6) and t = [ n−3 2 ] we have the following.Theorem. N\\G ss 2,n (L 2 ) has a stratification t i=0 C i where C 0 = S m+1 \P(h m ), and C i = S i+m+1 \V i+m .

Torus quotients of Richardson varieties in the Grassmannian

“…The following proposition was first proved by Kannan and Sardar, [8]. A simpler proof was given in [2].…”

“…Here w ′ = (4,5,8,9). Since w ′ > w 4,9 , X(w ′ ) contains semistable points and hence the quotient space T \ \X(w) ss T (L(9ω 4 )) is not smooth (using 3.2).…”

“…It is known from the work of Kumar [9] and also the work of Kannan and Sardar [8] that the line bundle L(dω r ) descends to the the GIT quotient T \ \X(w) ss T (L(dω r )) precisely when n|d.…”

“…Kannan and Sardar [8] showed that there is a unique minimal Schubert Variety X(w r,n ) in G r,n admitting semistable points with respect to the line bundle L(nω r ). We denote by (a 1 , a 2 , .…”

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

“…In [4] and [5], the parabolic subgroups Q of G containing B for which there exists an ample line bundle L on G/Q such that the semistable points (G/Q) ss T (L) are the same as the stable points (G/Q) s T (L). In [7], when Q is a maximal parabolic subgroup of G and L = L , where is a minuscule dominant weight, it is shown that there exists unique minimal dimensional Schubert variety X(w) admitting semistable points with respect to L.…”