LS Algebras, Valuations and Schubert Varieties

Abstract

In this paper, we propose an algebraic approach via Lakshmibai-Seshadri (LS) algebras to establish a link between standard monomial theories, Newton-Okounkov bodies and valuations. This is applied to Schubert varieties, where this approach is compatible with the one using Seshadri stratifications in [4], showing that LS paths encode vanishing multiplicities with respect to the web of Schubert varieties.

Seshadri stratifications and Schubert varieties: a geometric construction of a standard monomial theory

“…LS-algebras and valuations. The second step of our project (see [10]) showed that an LS-algebra may be defined in terms of valuations and a quasi-valuation as well. Also in this case one recovers the generators of the algebra via valuations having at most one-dimensional leaves, the quasi-valuation is additive on standard monomials and the straightening relations have a clear interpretation in terms of the quasi-valuation.…”

“…See also Conclusion 5.5 for connections with the present formulation. For the details with respect to the following formulation see [4], [6] and [10].…”