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 to characterize geometrical properties of Schubert varieties. Their results are very closely related to work of Haiman, Ryan and Wolper, but Lakshmibai-Sandhya were the first to use that language exactly. Pattern avoidance in permutations was used historically by Knuth, Pratt, Tarjan, and others in the 1960's and 1970's to characterize sorting algorithms in computer science. Lascoux and Schützenberger also used pattern avoidance to characterize vexillary permutations in the 1980's. Now, there are many geometrical properties of Schubert varieties that use pattern avoidance as a method for characterization including Gorenstein, factorial, local complete intersections, and properties of Kazhdan-Lusztig polynomials. These are what we call consequences of the Lakshmibai-Sandhya theorem. We survey the many beautiful results, generalizations, and remaining open problems in this area. We highlight the advantages of using pattern avoidance characterizations in terms of linear time algorithms and the ease of access to the literature via Tenner's Database of Permutation Pattern Avoidance. This survey is based on lectures by the second author at Osaka, Japan 2012 for the Summer School of the Mathematical Society of Japan based on the topic of Schubert calculus.

For a reductive group $G$, the products of projective varieties homogeneous under $G$ that are spherical for the diagonal action of $G$ have been classified by Stembridge. We consider the $B$-orbit closures in these spherical varieties and prove that under some mild restrictions they are normal, Cohen-Macaulay and have a rational resolution.

This paper reviews the moment graph technique that allows to translate certain representation theoretic problems into geometric ones. For simplicity we restrict ourselves to the case of semisimple complex Lie algebras. In particular, we show how the original Kazhdan–Lusztig conjecture on the characters of irreducible highest weight representations can be translated into a multiplicity problem for parity sheaves on the (Langlands dual) flag variety.

The notes are aimed at a reasonably self-contained introduction to the theory of Schubert polynomials (in a wider sense) for all the classical Lie types in the setting of torus equivariant cohomology. As a powerful combinatorial device, we use the restriction maps to the set of torus fixed points throughout the lectures.

We study the moduli space of twisted quasimaps from a fixed smooth projective curve to a Nakajima's quiver variety and the moduli space of $\delta$-stable framed twisted quiver bundles with moment map relations. We show that they carry symmetric obstruction theories and when $\delta$ is large enough, they exactly coincide. These results generalize works by D.E. Diaconescu [12] about the ADHM quiver, in the framework of the quasimap theory of I. Ciocan-Fontanine, D. Maulik and the author [8, 9].

We define convex-geometric counterparts of divided difference (or Demazure) operators from the Schubert calculus and representation theory. These operators are used to construct inductively polytopes that capture Demazure characters of representations of reductive groups. In particular, Gelfand–Zetlin polytopes and twisted cubes of Grossberg–Karshon are obtained in a uniform way.

These are notes for four lectures given at the Osaka summer school on Schubert calculus in 2012, presenting the geometry from the unpublished arXiv:1008.4302 giving an extension of the puzzle rule for Schubert calculus to equivariant $K$-theory, while eliding some of the combinatorial detail. In particular, §3 includes background material on equivariant cohomology and $K$-theory.

Since that school, I have extended the results to arbitrary interval positroid varieties (not just those arising in Vakil's geometric Littlewood-Richardson rule), in the preprint [Kn2].

This mostly expository article explores recent developments in the relations between the three objects in the title from an algebro-combinatorial perspective.

We prove a formula for Whittaker functions of a real semisimple group as an integral over a geometric crystal in the sense of Berenstein-Kazhdan. We explain the connections of this formula to the program of mirror symmetry of flag varieties developed by Givental and Rietsch; in particular, the integral formula proves the equivariant version of Rietsch's mirror symmetry conjecture. We also explain the idea that Whittaker functions should be thought of as geometric analogues of irreducible characters of finite-dimensional representations.

The Ehresmann-Bruhat order on the symmetric group satisfies a symmetry property that we generalize to a directed graph with permutations as vertices, labeling edges with tableaux of a given shape.

We give an explicit description of the image of a quantum LS path, regarded as a rational path, under the action of root operators, and show that the set of quantum LS paths is stable under the action of the root operators. As a by-product, we obtain a new proof of the fact that a projected level-zero LS path is just a quantum LS path.

Many aspects of Schubert calculus are easily modeled on a computer. This enables large-scale experimentation to investigate subtle and ill-understood phenomena in the Schubert calculus. A well-known web of conjectures and results in the real Schubert calculus has been inspired by this continuing experimentation. A similarly rich story concerning intrinsic structure, or Galois groups, of Schubert problems is also beginning to emerge from experimentation. This showcases new possibilities for the use of computers in mathematical research.

In this paper, we study the generalized (co)homology Hopf algebras of the loop spaces on the infinite classical groups, generalizing the work due to Kono-Kozima and Clarke. We shall give a description of these Hopf algebras in terms of symmetric functions. Based on topological considerations in the first half of this paper, we then introduce a universal analogue of the factorial Schur $P$- and $Q$-functions due to Ivanov and Ikeda-Naruse. We investigate various properties of these functions such as the cancellation property, which we call the $\mathbb{L}$-supersymmetric property, the factorization property, and the vanishing property. We prove that the universal analogue of the Schur $P$-functions form a formal basis for the ring of symmetric functions with the $\mathbb{L}$-supersymmetric property. By using the universal analogue of the Cauchy identity, we then define the dual universal Schur $P$- and $Q$-functions. We describe the duality of these functions in terms of Hopf algebras.

We describe the positivity of Thom polynomials of singularities of maps, Lagrangian Thom polynomials and Legendrian Thom polynomials. We show that these positivities come from Schubert calculus.

This paper is a survey on a joint work with K. Sorlin concerning the theory of character sheaves on the exotic symmetric space. After explaining the historical background, we introduce character sheaves on this variety. By using those character sheaves, we show that modified Kostka polynomials, indexed by a pair of double partitions, can be interpreted in terms of the intersection cohomology of the orbits in the exotic nilpotent cone, as conjectured in Achar-Henderson.

We study “cominuscule tableau combinatorics” by generalizing constructions of M. Haiman, S. Fomin and M.-P. Schützenberger. In particular, we extend the dual equivalence ideas of [Haiman, 1992] to reformulate the generalized Littlewood-Richardson rule for cominuscule $G/P$ Schubert calculus from [Thomas-Yong, 2006]. We apply dual equivalence to give an alternative and independent proof of the jeu de taquin results of [Proctor, 2004] needed in our earlier work. We also extend Fomin's growth diagram description of jeu de taquin; the inherent symmetry of these diagrams leads to a generalization of Schützenberger's evacuation involution. Finally, these results are applied to give an cominuscule extension of the carton rule of [Thomas-Yong, 2008].