Tableau formulas for skew Schubert polynomials

Schubert Polynomials and the Nilcoxeter Algebra

Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for the construction of Schubert polynomials; we also present simplifications in some of the existing approaches to this area. We designate certain line diagrams for permutations known as rc-graphs as the main structure. The other structures in the literature we study include: semistandard Young tableaux, Kohnert diagrams, and balanced labelings of the diagram of a permutation. The main tools in our investigation are certain operations on rc-graphs, which correspond to the coplactic operations on tableaux, and thus define a crystal graph structure on rc-graphs; a new definition of these operations is presented. One application of these operations is a straightforward, purely combinatorial proof of a recent formula (due to Buch, Kresch, Tamvakis, and Yong), which expresses Schubert polynomials in terms of products of Schur polynomials. In spite of the fact that it refers to many objects and results related to them, the paper is mostly self-contained.

We use the Pieri and Giambelli formulas of Buch et al. (Invent Math 178:345–405, 2009; J Reine Angew, 2013) and the calculus of raising operators developed in Buch et al. (A Giambelli formula for isotropic Grassmannians, arXiv:0811.2781, 2008) and Tamvakis (J Reine Angew Math 652, 207–244, 2011) to prove a tableau formula for the eta polynomials of Buch et al. (J Reine Angew, 2013) and the Stanley symmetric functions which correspond to Grassmannian elements of the Weyl group $$\widetilde{W}_n$$ of type $$\text {D}_n$$ . We define the skew elements of $$\widetilde{W}_n$$ and exhibit a bijection between the set of reduced words for any skew $$w\in \widetilde{W}_n$$ and a set of certain standard typed tableaux on a skew shape $$\lambda /\mu $$ associated to $$w$$ .

In 1988, Macdonald introduced two remarkable families of polynomials which bear his name. The first family is the symmetric Macdonald polynomials, which generalize the irreducible characters of semisimple Lie algebras. There are two-well known combinatorial formulas for the symmetric Macdonald polynomials — the Haglund-Haiman-Loehr formula, expressed in terms of certain tableaux but only defined in Lie type A, and the Ram-Yip formula, expressed in terms of alcove paths and defined in all Lie types. The connection between these two formulas has been established by Lenart.

We investigate the symmetric Dunkl-classical orthogonal polynomials by using a new approach in connection with the Dunkl operator. The main aim of this technique is first and foremost to determine the recurrence coefficients. We establish the existence and uniqueness of symmetric Dunkl-classical orthogonal polynomials, and confirm that the two families of generalized Hermite orthogonal polynomials and Gegenbauer orthogonal polynomials are the only ones belonging to this class. Two apparently new characterizations in a more general setting are given. This paper complements earlier work of Ben Cheikh and Gaied [Characterization of the Dunkl-classical symmetric orthogonal polynomials. Appl Math Comput. 2007;187:105–114. doi: 10.1016/j.amc.2006.08.108].

The Hi-Jack symmetric polynomials, which are associated with the simultaneous\neigenstates for the first and second conserved operators of the quantum\nCalogero model, are studied. Using the algebraic properties of the Dunkl\noperators for the model, we derive the Rodrigues formula for the Hi-Jack\nsymmetric polynomials. Some properties of the Hi-Jack polynomials and the\nrelationships with the Jack symmetric polynomials and with the basis given by\nthe QISM approach are presented. The Hi-Jack symmetric polynomials are strong\ncandidates for the orthogonal basis of the quantum Calogero model.\n

We study the explicit formula of Lusztig’s integral forms of the level one quantum affine algebraUq(sl^2)U_q(\widehat {sl}_2)in the endomorphism ring of symmetric functions in infinitely many variables tensored with the group algebra ofZ\mathbb Z. Schur functions are realized as certain orthonormal basis vectors in the vertex representation associated to the standard Heisenberg algebra. In this picture the Littlewood-Richardson rule is expressed by integral formulas, and is used to define the action of Lusztig’sZ[q,q−1]\mathbb Z[q, q^{-1}]-form ofUq(sl^2)U_q(\widehat {sl}_2)on Schur polynomials. As a result theZ[q,q−1]\mathbb Z[q, q^{-1}]-lattice of Schur functions tensored with the group algebra contains Lusztig’s integral lattice.

Bumpless pipe dreams and alternating sign matrices

Orthogonal intertwiners for infinite particle systems in the continuum

Matrix valued Hermite polynomials, Burchnall formulas and non-abelian Toda lattice

Quadrature formulas for Bessel polynomials

Companion orthogonal polynomials: some applications

Characterization of the Dunkl-classical symmetric orthogonal polynomials

We address two fundamental questions in the representation theory of affine Hecke algebras of classical types. One is an inductive algorithm to compute characters of tempered modules, and the other is the determination of the constants in the formal degrees of discrete series (in the form conjectured by Reeder (J. Reine Angew. Math. 520:37–93, 2000)). The former is completely different from the Lusztig-Shoji algorithm (Shoji in Invent. Math. 74:239–267, 1983; Lusztig in Ann. Math. 131:355–408, 1990), and it is more effective in a number of cases. The main idea in our proof is to introduce a new family of representations which behave like tempered modules, but for which it is easier to analyze the effect of parameter specializations. Our proof also requires a comparison of the C ∗-theoretic results of Opdam, Delorme, Slooten, Solleveld (J. Inst. Math. Jussieu 3:531–648, 2004; arXiv:0909.1227 ; Int. Math. Res. Not., 2008; Adv. Math. 220:1549–1601, 2009; Acta Math. 205:105–187, 2010), and the geometric construction from Kato (Duke Math. J. 148:305–371, 2009; Am. J. Math. 133:518–553, 2011), Ciubotaru and Kato (Adv. Math. 226:1538–1590, 2011).

In this paper, we obtain the ladder operators and associated compatibility conditions for types I and II multiple orthogonal polynomials. These ladder equations extend known results for orthogonal polynomials and can be used to derive the differential equations satisfied by multiple orthogonal polynomials. Our approach is based on Riemann–Hilbert problems and the Christoffel–Darboux formula for multiple orthogonal polynomials, and the nearest-neighbor recurrence relations. As an illustration, we give several explicit examples involving multiple Hermite and Laguerre polynomials, and multiple orthogonal polynomials with exponential weights and cubic potentials.