56 research outputs found
Staircase diagrams and the enumeration of smooth Schubert varieties
International audienceIn this extended abstract, we give a complete description and enumeration of smooth and rationally smooth Schubert varieties in finite type. In particular, we show that rationally smooth Schubert varieties are in bijection with a new combinatorial data structure called staircase diagrams
Divisor labelling of staircase diagrams and fiber bundle structures on Schubert varieties
Let Gr(, ) denote the Grassmannian of -dimensional subspaces of the -dimensional vector space ℂⁿ over the field of complex numbers. Each Gr(, ) contains a unique codimension 1 Schubert subvariety called the Schubert divisor of the Grassmannian. In this project, we will discuss the correspondence between the set of permutations avoiding the patterns 3412, 52341, 52431, and 53241, and the set of Schubert varieties in the complete flag variety which are iterated fiber bundles of Grassmannians or Grassmannian Schubert divisors. Using this geometrical structure, we calculate the generating function that enumerates the permutations avoiding these patterns
Parabolic double cosets in Coxeter groups
International audienceParabolic subgroups WI of Coxeter systems (W,S) and their ordinary and double cosets W/WI and WI\W/WJ appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets wWI , for I ⊆ S, forms the Coxeter complex of W , and is well-studied. In this extended abstract, we look at a less studied object: the set of all double cosets WIwWJ for I,J ⊆ S. Each double coset can be presented by many different triples (I, w, J). We describe what we call the lex-minimal presentation and prove that there exists a unique such choice for each double coset. Lex-minimal presentations can be enumerated via a finite automaton depending on the Coxeter graph for (W, S). In particular, we present a formula for the number of parabolic double cosets with a fixed minimal element when W is the symmetric group Sn. In that case, parabolic subgroups are also known as Young subgroups. Our formula is almost always linear time computable in n, and the formula can be generalized to any Coxeter group
K-classes of Brill-Noether loci and a determinantal formula
We prove a determinantal formula for the K-theory class of certain degeneracy
loci, and apply it to compute the Euler characteristic of the structure sheaf
of the Brill-Noether locus of linear series with special vanishing at marked
points. When the Brill-Noether number is zero, we recover the
Castelnuovo formula for the number of special linear series on a general curve;
when , we recover the formulas of Eisenbud-Harris, Pirola, and
Chan-L\'opez-Pflueger-Teixidor for the arithmetic genus of a Brill-Noether
curve of special divisors.
Our degeneracy locus formula also specializes to new determinantal
expressions for the double Grothendieck polynomials corresponding to
321-avoiding permutations, and gives double versions of the flagged skew
Grothendieck polynomials recently introduced by Matsumura. Our result extends
the formula of Billey-Jockusch-Stanley expressing Schubert polynomials for
321-avoiding permutations as generating functions for skew tableaux.Comment: 31 pages; v2: stronger Theorem C, and improved expositio
Self-dual intervals in the Bruhat order
Bj\"orner-Ekedahl prove that general intervals in Bruhat order are
"top-heavy", with at least as many elements in the -th corank as the -th
rank. Well-known results of Carrell and of Lakshmibai-Sandhya give the equality
case: is rank-symmetric if and only if the permutation avoids the
patterns and and these are exactly those such that the
Schubert variety is smooth.
In this paper we study the finer structure of rank-symmetric intervals
, beyond their rank functions. In particular, we show that these
intervals are still "top-heavy" if one counts cover relations between different
ranks. The equality case in this setting occurs when is self-dual as a
poset; we characterize these by pattern avoidance and in several other
ways.Comment: 22 pages; v2: minor edits and journal referenc
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