arXiv subjects
Publications and source records attributed to Peter Littelmann.
Using a bijection between the lattice points in a lecture hall polytope and Lakshimibai-Seshadri (L-S) paths, we prove the Koszul property of lecture hall polytopes in complete generality, and give new proofs for their Integral Decomposition Property and for a criterion on Gorenstein property.
We define and study the higher rank GKZ-fans of point configurations, where the rank one cases coincide with the usual GKZ-fans. A point in a higher rank GKZ-fan is then used to construct higher rank quasi-valuations to degenerate the toric variety associated to the point configuration flatly to a reduced union of toric varieties. Such a union encodes the polytopal subdivision arising from the point in the higher rank GKZ-fan.
We apply the theory of Seshadri stratifications to embedded toric varieties $X_P\subseteq \mathbb P(V)$ associated with a normal lattice polytope $P$. The approach presented here is purely combinatorial and completely independent of \cite{CFL}. In particular, we get a close connection between a certain class of triangulations of the polytope $P$, Seshadri stratifications of $X_P$ arising from torus orbit closures, and the associated degenerate semi-toric varieties. In the last section we show that the approach here and the one in \cite{CFL} produce the same quasi-valuations and hence the same degenerations of $X_P$.
The goal of the paper is twofold: on one side it provides an order structure on the set of all maximal chains in the Bruhat poset of Schubert varieties in a Grassmann variety; on the other hand, using this order structure, it works out explicit formulae for the valuation and the Newton-Okounkov body associated to each maximal chain appearing in the framework of Seshadri stratification.
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 [32] is compatible with this new strategy.
A standard monomial theory for Schubert varieties is constructed exploiting (1) the geometry of the Seshadri stratifications of Schubert varieties by their Schubert subvarieties and (2) the combinatorial LS-path character formula for Demazure modules. The general theory of Seshadri stratifications is improved by using arbitrary linearization of the partial order and by weakening the definition of balanced stratification.
The existence of a Seshadri stratification on an embedded projective variety provides a flat degeneration of the variety to a union of projective toric varieties, called a semi-toric variety. Such a stratification is said to be normal when each irreducible component of the semi-toric variety is a normal toric variety. In this case, we show that a Gr\"obner basis of the defining ideal of the semi-toric variety can be lifted to define the embedded projective variety. Applications to Koszul and Gorenstein properties are discussed. Relations between LS-algebras and certain Seshadri stratifications are studied.
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 by the same authors (arXiv:2112.03776), showing that LS paths encode vanishing multiplicities with respect to the web of Schubert varieties.
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.
The geometric Satake correspondence can be regarded as a geometric construction of the rational representations of a complex connected reductive group G. In their study of this correspondence, Mirković and Vilonen introduced algebraic cycles that provide a linear basis in each irreducible representation. Generalizing this construction, Goncharov and Shen define a linear basis in each tensor product of irreducible representations. We investigate these bases and show that they share many properties with the dual canonical bases of Lusztig.
We give a uniform proof for the conjectured Gromov width of coadjoint orbits of all compact connected simple Lie groups, by analyzing simplices in 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.
We present a new approach to construct $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, combining ideas coming from the theory of Newton-Okounkov bodies with ideas originally stemming from PBW-filtrations. For each pair $(S,>)$ consisting of a birational sequence and a monomial order, we attach to the affine variety $G/\hskip -3.5pt/U$ a monoid $Γ=Γ(S,>)$. As a side effect we get a vector space basis $\mathbb B_Γ$ of $\mathbb C[G/\hskip -3.5pt/U]$, the elements being indexed by $Γ$. The basis $\mathbb B_Γ$ has multiplicative properties very similar to those of the dual canonical basis. This makes it possible to transfer the methods of Alexeev and Brion \cite{AB} to this more general setting, once one knows that the monoid $Γ$ is finitely generated and saturated.
Following the historical track in pursuing $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, we explain how powerful tools in algebraic geometry and representation theory, such as canonical bases, Newton-Okounkov bodies, PBW-filtrations and cluster algebras come to push the subject forward.
We consider the PBW filtrations over the integers of the irreducible highest weight modules in type A and C. We show that the associated graded modules can be realized as Demazure modules for group schemes of the same type and doubled rank. We deduce that the corresponding degenerate flag varieties are isomorphic to Schubert varieties in any characteristic.
We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis. In the favourable case a basis of the module is parameterized by the lattice points of a normal polytope. The filtrations induce flat degenerations of the corresponding flag variety to its abelianized version and to a toric variety, the special fibres of the degenerations being projectively normal and arithmetically Cohen-Macaulay. The polytope itself can be recovered as a Newton-Okounkov body. We conclude the paper by giving classes of examples for favourable modules.
We study the category of finite--dimensional bi--graded representations of toroidal current algebras associated to finite--dimensional complex simple Lie algebras. Using the theory of graded representations for current algebras, we construct in different ways objects in that category and prove them to be isomorphic. As a consequence we obtain generators and relations for certain types of fusion products including the $N$--fold fusion product of $V(λ)$. This result shows that the fusion product of these types is independent of the chosen parameters, proving a special case of a conjecture by Feigin and Loktev. Moreover, we prove a conjecture by Chari, Fourier and Sagaki on truncated Weyl modules for certain classes of dominant integral weights and show that they are realizable as fusion products. In the last section we consider the case $\mathfrak{g}=\mathfrak{sl}_2$ and compute a PBW type basis for truncated Weyl modules of the associated current algebra.
We give a geometric interpretation of the Knuth equivalence relations in terms of the affine Graß mann variety. The Young tableaux are seen as sequences of coweights, called galleries. We show that to any gallery corresponds a Mirković-Vilonen cycle and that two galleries are equivalent if, and only if, their associated MV cycles are equal. Words are naturally identified to some galleries. So, as a corollary, we obtain that two words are Knuth equivalent if, and only if, their associated MV cycles are equal.