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Toric degeneration of Schubert varieties and Gelfand–Tsetlin polytopes. (English) Zbl 1084.14049
This paper connects two degenerations related to the manifold \(F_n\) of complete flags in \({\mathbb C}^n\). N. Gonciulea and V. Lakshmibai [Transform. Groups 1, No. 3, 215–248 (1996; Zbl 0909.14028)] used standard monomial theory to construct a flat sagbi degeneration of \(F_n\) into the toric variety of the Gelfand-Tsetlin polytope, and recently A. Knutson and E. Miller [Ann. Math. (2) 161, No. 3, 215–248 (2005; Zbl 1089.14007)] constructed Gröbner degenerations of matrix Schubert varieties into linear spaces corresponding to monomials in double Schubert polynmials. The flag variety is the geometric invariant theory (GIT) quotient of the space \(M_n\) of \(n\) by \(n\) matrices by the Borel group \(B\) of lower triangular matrices. A matrix Schubert variety is an inverse image of a Schubert variety under this quotient. The main result in the paper under review is that this GIT quotient extends to the degenerations. The sagbi degeneration is a GIT quotient of the Gröbner degeneration.\smallskip The nature of this GIT quotient is quite interesting. The authors exhibit an action of the Borel group \(B\) on the product \(M_n\times{\mathbb C}\) of \(M_n\) with the complex line, so that the GIT quotient \(B\backslash\backslash(M_n\times{\mathbb C})\) remains fibred over \({\mathbb C}\) and is the total space of the sagbi degeneration. In this GIT quotient, the total space of the Gröbner degeneration (as in Knutson and Miller) of a matrix Schubert variety in \(M_n\times{\mathbb C}\) covers the total space of the Lakshmibai-Gonciulea degeneration of the corresponding Schubert variety. At the degenerate point, the matrix Schubert variety has become a union of coordinate planes, each of which covers a component of the sagbi degeneration of the Schubert variety indexed by a face of the Gelfand-Tsetlin polytope.
The authors use this to identify which faces of the Gelfand-Tsetlin polytope occur in a given degenerate Schubert variety, and to give a simple explanation of the classical Gelfand-Tsetlin decomposition of an irreducible polynomial representation of \(\text{GL}_n\) into one-dimensional weight spaces; in the degeneration, sections of a line bundle over \(F_n\) become sections of the defining line bundle on the toric variety for the Gelfand-Tsetlin polytope.

14M15 Grassmannians, Schubert varieties, flag manifolds
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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