Standard Monomial Theory Research Articles

Standard monomials and invariant theory for arc spaces III: Special linear group

This is the third in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field K, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the special linear group. This is more subtle than the results for the general linear and symplectic groups obtained in the first two papers because the arc space of the corresponding affine quotients can be nonreduced.

Standard monomials and invariant theory of arc spaces II: Symplectic group

This is the second in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field K K , we construct a standard monomial basis for the arc space of the Pfaffian variety over K K . As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the symplectic group.

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

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

Seshadri stratifications and standard monomial theory

We introduce the notion of a Seshadri stratification on an embedded projective variety. Such a structure enables us to construct a Newton-Okounkov simplicial complex and a flat degeneration of the projective variety into a union of toric varieties. We show that the Seshadri stratification provides a geometric setup for a standard monomial theory. In this framework, Lakshmibai-Seshadri paths for Schubert varieties get a geometric interpretation as successive vanishing orders of regular functions.

Standard monomials and invariant theory for arc spaces I: General linear group

This is the first in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field [Formula: see text], we construct a standard monomial basis for the arc space of the determinantal variety over [Formula: see text]. As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the general linear group.

Seshadri stratification for Schubert varieties and standard monomial theory

The theory of Seshadri stratifications has been developed by the authors with the intention to build up a new geometric approach towards a standard monomial theory for embedded projective varieties with certain nice properties. In this article, we investigate the Seshadri stratification on a Schubert variety arising from its Schubert subvarieties. We show that the standard monomial theory developed in Littelmann (J. Am. Math. Soc. 11(3) (1998) 551–567) is compatible with this new strategy.

Chevalley formula for anti-dominant weights in the equivariant K-theory of semi-infinite flag manifolds

We prove a Chevalley formula for anti-dominant weights in the torus-equivariant K-group of semi-infinite flag manifolds, which is described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or equivalently, quantum Lakshmibai-Seshadri paths); in contrast to the Chevalley formula for dominant weights in our previous paper [17], the formula for anti-dominant weights has a significant finiteness property. Based on geometric results established in [17], our proof is representation-theoretic, and the Chevalley formula for anti-dominant weights follows from a certain identity for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra; in the proof of this identity, we make use of the (combinatorial) standard monomial theory for semi-infinite Lakshmibai-Seshadri paths, and also a string property of Demazure-like subsets of the set of semi-infinite Lakshmibai-Seshadri paths of a fixed shape, which gives an explicit realization of the crystal basis of a level-zero extremal weight module.

Standard monomial theory and toric degenerations of Richardson varieties in the Grassmannian

Richardson varieties are obtained as intersections of Schubert and opposite Schubert varieties. We provide a new family of toric degenerations of Richardson varieties inside Grassmannians by studying Gröbner degenerations of their corresponding ideals. These degenerations are parametrised by block diagonal matching fields in the sense of Sturmfels [33]. We associate a weight vector to each block diagonal matching field and study its corresponding initial ideal. In particular, we characterise when such ideals are toric, hence providing a family of toric degenerations for Richardson varieties. Given a Richardson variety $$X_{w}^v$$ and a weight vector $$\mathbf{w}_\ell $$ arising from a matching field, we consider two ideals: an ideal $$G_{k,n,\ell }|_w^v$$ obtained by restricting the initial of the Plücker ideal to a smaller polynomial ring, and a toric ideal defined as the kernel of a monomial map $$\phi _\ell |_w^v$$ . We first characterise the monomial-free ideals of form $$G_{k,n,\ell }|_w^v$$ . Then we construct a family of tableaux in bijection with semi–standard Young tableaux which leads to a monomial basis for the corresponding quotient ring. Finally, we prove that when $$G_{k,n,\ell }|_w^v$$ is monomial-free and the initial ideal in $$_{\mathbf{w}_\ell }(I(X_w^v))$$ is quadratically generated, then all three ideals in $$_{\mathbf{w}_\ell }(I(X_w^v))$$ , $$G_{k,n,\ell }|_w^v$$ and ker $$(\phi _\ell |_w^v)$$ coincide, and provide a toric degeneration of $$X_w^v$$ .

Standard monomial theory and toric degenerations of Schubert varieties from matching field tableaux

We study Gröbner degenerations of Schubert varieties inside flag varieties. We consider toric degenerations of flag varieties induced by matching fields and semi-standard Young tableaux. We describe an analogue of matching field ideals for Schubert varieties inside the flag variety and give a complete characterization of toric ideals among them. We use a combinatorial approach to standard monomial theory to show that block diagonal matching fields give rise to toric degenerations. Our methods and results use the combinatorics of permutations associated to Schubert varieties, matching fields and their corresponding tableaux.

Equivariant K -theory of semi-infinite flag manifolds and the Pieri–Chevalley formula

We propose a definition of equivariant (with respect to an Iwahori subgroup) $K$-theory of the formal power series model $\mathbf{Q}_{G}$ of semi-infinite flag manifold and prove the Pieri-Chevalley formula, which describes the product, in the $K$-theory of $\mathbf{Q}_{G}$, of the structure sheaf of a semi-infinite Schubert variety with a line bundle (associated to a dominant integral weight) over $\mathbf{Q}_{G}$. In order to achieve this, we provide a number of fundamental results on $\mathbf{Q}_{G}$ and its Schubert subvarieties including the Borel-Weil-Bott theory, whose special case is conjectured in [A. Braverman and M. Finkelberg, Weyl modules and $q$-Whittaker functions, Math. Ann. 359 (2014), 45--59]. One more ingredient of this paper besides the geometric results above is (a combinatorial version of) standard monomial theory for level-zero extremal weight modules over quantum affine algebras, which is described in terms of semi-infinite Lakshmibai-Seshadri paths. In fact, in our Pieri-Chevalley formula, the positivity of structure coefficients is proved by giving an explicit representation-theoretic meaning through semi-infinite Lakshmibai-Seshadri paths.

Tensor product decomposition theorem for quantum Lakshmibai-Seshadri paths and standard monomial theory for semi-infinite Lakshmibai-Seshadri paths

Let λ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let QLS(λ) denote the set of quantum Lakshmibai-Seshadri (QLS) paths of shape λ. For an element w of a finite Weyl group W, the specializations at t=0 and t=∞ of the nonsymmetric Macdonald polynomial Ewλ(q,t) are explicitly described in terms of QLS paths of shape λ and the degree function defined on them. Also, for (level-zero) dominant integral weights λ, μ, we have an isomorphism Θ:QLS(λ+μ)→QLS(λ)⊗QLS(μ) of crystals. In this paper, we study the behavior of the degree function under the isomorphism Θ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules.

From standard monomial theory to semi-toric degenerations via Newton–Okounkov bodies

The Hodge algebra structures on the homogeneous coordinate rings of Grassmann varieties provide semi-toric degenerations of these varieties. In this paper we construct these semi-toric degenerations using quasi-valuations and triangulations of Newton-Okounkov bodies.

Standard Monomial Theory for Wonderful Varieties

A general setting for a standard monomial theory on a multiset is introduced and applied to the Cox ring of a wonderful variety. This gives a degeneration result of the Cox ring to a multicone over a partial flag variety. Further, we deduce that the Cox ring has rational singularities.

Standard Monomial Theory of RR Varieties

We construct the RR varieties as the fiber products of Bott-Samelson varieties over Richardson varieties. We study their homogeneous coordinate rings and standard monomial theory.

Multiplicities of classical varieties

The $j$-multiplicity plays an important role in the intersection theory of St\"uckrad-Vogel cycles, while recent developments confirm the connections between the $\epsilon$-multiplicity and equisingularity theory. In this paper we establish, under some constraints, a relationship between the $j$-multiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute the $j$-multiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute the $j$- and $\epsilon$-multiplicity of ideals defining determinantal varieties: The found quantities are integrals which, quite surprisingly, are central in random matrix theory.

PLÜCKER RELATIONS AND SPHERICAL VARIETIES: APPLICATION TO MODEL VARIETIES

A general framework for the reduction of the equations defining classes of spherical varieties to (possibly infinite-dimensional) grassmannians is proposed. This is applied to model varieties of types A, B and C; in particular, a standard monomial theory for these varieties is presented.

Standard monomial theory for desingularized Richardson varieties in the flag variety GL(n)/B

We consider a desingularization Γ of a Richardson variety in the variety of complete flags, obtained as a fibre of a projection from a certain Bott-Samelson variety Z. For any projective embedding of Z via a complete linear system, we construct a basis of the homogeneous coordinate ring of Γ inside Z, indexed by combinatorial objects which we call w 0 -standard tableaux.

Pfaffians and Shuffling Relations for the Spin Module

We present explicit formulas for a set of generators of the ideal of relations among the pfaffians of the principal minors of the antisymmetric matrices of fixed dimension. These formulas have an interpretation in terms of the standard monomial theory for the spin module of orthogonal groups.

A non-crossing standard monomial theory

The second author has introduced non-crossing tableaux, objects whose non-nesting analogues are semi-standard Young tableaux. We relate non-crossing tableaux to Gelfand–Tsetlin patterns and develop the non-crossing analogue of standard monomial theory. Leclerc and Zelevinsky's weakly separated sets are special cases of non-crossing tableaux, and we suggest that non-crossing tableaux may help illuminate the theory of weakly separated sets.

The nullcone in the multi-vector representation of the symplectic group and related combinatorics

We study the nullcone in the multi-vector representation of the symplectic group with respect to a joint action of the general linear group and the symplectic group. By extracting an algebra over a distributive lattice structure from the coordinate ring of the nullcone, we describe a toric degeneration and standard monomial theory of the nullcone in terms of double tableaux and integral points in a convex polyhedral cone.