# Permutation patterns, Stanley symmetric functions, and generalized Specht modules

@article{Billey2014PermutationPS, title={Permutation patterns, Stanley symmetric functions, and generalized Specht modules}, author={Sara C. Billey and Brendan Pawlowski}, journal={J. Comb. Theory, Ser. A}, year={2014}, volume={127}, pages={85-120} }

Abstract Generalizing the notion of a vexillary permutation, we introduce a filtration of S ∞ by the number of terms in the Stanley symmetric function, with the kth filtration level called the k-vexillary permutations. We show that for each k, the k-vexillary permutations are characterized by avoiding a finite set of patterns. A key step is the construction of a Specht series, in the sense of James and Peel, for the Specht module associated with the diagram of a permutation. As a corollary, we…

## 14 Citations

Consequences of the Lakshmibai-Sandhya Theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry

- Mathematics
- 2014

In 1990, Lakshmibai and Sandhya published a characterization of singular Schubert varieties in flag manifolds using the notion of pattern avoidance. This was the first time pattern avoidance was used…

Cohomology Classes of Interval Positroid Varieties and a Conjecture of Liu

- Computer Science, MathematicsElectron. J. Comb.
- 2018

It is shown that for the diagram variety of a permutation diagram, Liu's conjecture that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes.

Cohomology classes of rank varieties and a counterexample to a conjecture of Liu

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- 2015

To each finite subset of a discrete grid $\mathbb{N}×\mathbb{N}$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group…

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To each finite subset of a discrete grid $\mathbb{N} \times \mathbb{N}$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric…

Combinatorial generation via permutation languages. I. Fundamentals

- Computer Science, MathematicsTransactions of the American Mathematical Society
- 2020

This work presents a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations, and obtains four classical Gray codes for permutation, bitstrings, binary trees and set partitions as special cases.

A representation-theoretic interpretation of positroid classes

- Mathematics
- 2016

A positroid is the matroid of a real matrix with nonnegative maximal minors, a positroid variety is the closure of the locus of points in a complex Grassmannian whose matroid is a fixed positroid,…

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- Mathematics
- 2015

We develop the technique of reduced word manipulation to give a range of results concerning reduced words and permutations more generally. We prove a broad connection between pattern containment and…

A natural generalization of Balanced Tableaux

- Mathematics
- 2016

We introduce the notion of ``type'' of a tableau, that allows us to define new families of tableaux including both balanced and standard Young tableaux. We use these new objects to describe the set…

Maximizing the Edelman-Greene statistic

- Mathematics
- 2019

The $\textit{Edelman-Greene statistic}$ of S. Billey-B. Pawlowski measures the "shortness" of the Schur expansion of a Stanley symmetric function. We show that the maximum value of this statistic on…

Bumpless Pipedreams, Reduced Word Tableaux and Stanley Symmetric Functions

- Mathematics
- 2018

Lam, Lee and Shimozono introduced the structure of bumpless pipedreams in their study of back stable Schubert calculus. They found that a specific family of bumpless pipedreams, called EG-pipedreams,…

## References

SHOWING 1-10 OF 63 REFERENCES

Reduced decompositions and permutation patterns

- Mathematics
- 2005

Billey, Jockusch, and Stanley characterized 321-avoiding permutations by a property of their reduced decompositions. This paper generalizes that result with a detailed study of permutations via their…

Specht modules and Schubert varieties for general diagrams

- Mathematics
- 2010

The algebra of symmetric functions, the representation theory of the symmetric group, and the geometry of the Grassmannian are related to each other via Schur functions, Specht modules, and Schubert…

The X-Class and Almost-Increasing Permutations

- Mathematics
- 2007

In this paper we give a bijection between the class of permutations that can be drawn on an X-shape and a certain set of permutations that appears in Knuth [4] in connection to sorting algorithms. A…

Borel-Weil theorem for con guration varieties and Schur modules

- Mathematics
- 1998

We present a generalization of the classical Schur modules ofGL(n) exhibiting the same interplay among algebra, geometry, and combinatorics. A generalized Young diagramDis an arbitrary finite subset…

Degeneracy Loci, Pfaffians, and Vexillary Signed Permutations in Types B, C, and D

- Mathematics
- 2012

We define a notion of vexillary signed permutation in types B, C, and D, corresponding to natural degeneracy loci for vector bundles with symmetries of those types. We show that the classes of these…

Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. A
- 2012

It is shown that, for any k?0, the set of 1324-avoiding permutations with k inversions contains at least as many permutations of length n+1 as those of length eπ2/3?13, and that if this is true then the Stanley-Wilf limit for 1324 is at most eπ1/2?13.

Permutation patterns, Stanley symmetric functions, and the Edelman-Greene correspondence

- Mathematics
- 2013

Generalizing the notion of a vexillary permutation, we introduce a filtration of S1 by the number of Edelman-Greene tableaux of a permutation, and show that each filtration level is characterized by…

Percentage-Avoiding, Northwest Shapes and Peelable Tableaux

- Computer Science, MathematicsJ. Comb. Theory, Ser. A
- 1998

Three results are proved, giving an explicit combinatorial description for the multiplicities of irreducibles in the Specht and Schur modules of a %-avoiding shapeD, in terms of D-peelable tableaux, and three involutions on the set of peelable tableau which exhibit the symmetries of thesemultiplicities corresponding to three natural involutive operations.

Flags, Schubert polynomials, degeneracy loci, and determinantal formulas

- Mathematics
- 1992

Under appropriate conditions on the rank function r, which guarantee that, for generic h, f,(h) is irreducible, we prove a formula for the class [f,(h)] of this locus in the Chow or cohomology ring…

Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams

- Mathematics
- 2014

We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations…