Pieri Rule Research Articles

Equivariant Kazhdan-Lusztig polynomials of thagomizer matroids

The equivariant Kazhdan-Lusztig polynomial of a matroid was introduced by Gedeon, Proudfoot, and Young. Gedeon conjectured an explicit formula for the equivariant Kazhdan-Lusztig polynomials of thagomizer matroids with an action of symmetric groups. In this paper, we discover a new formula for these polynomials which is related to the equivariant Kazhdan-Lusztig polynomials of uniform matroids. Based on our new formula, we confirm Gedeon’s conjecture by the Pieri rule.

Formula omitted]-hypergeometric solutions of quantum differential equations, quantum Pieri rules, and Gamma theorem

We describe q-hypergeometric solutions of the equivariant quantum differential equations and the associated qKZ difference equations for the cotangent bundle T∗Fλ of a partial flag variety Fλ. These q-hypergeometric solutions manifest a Landau–Ginzburg mirror symmetry for the cotangent bundle. We formulate and prove Pieri rules for quantum equivariant cohomology of the cotangent bundle. Our Gamma theorem for T∗Fλsays that the leading term of the asymptotics of the q-hypergeometric solutions can be written as the equivariant Gamma class of the tangent bundle of T∗Fλ multiplied by the exponentials of the equivariant first Chern classes of the associated vector bundles. That statement is analogous to the statement of the gamma conjecture by B.Dubrovin and by S.Galkin, V.Golyshev, and H.Iritani, see also the Gamma theorem for Fλin Appendix B.

A geometric proof of an equivariant Pieri rule for flag manifolds

Abstract We use geometry to give a short proof of an equivariant Pieri rule in the classical flag manifold. This rule is due to Robinson, who gave an algebraic proof.

Pieri Rules for the Jack Polynomials in Superspace and the 6-Vertex Model

We present Pieri rules for the Jack polynomials in superspace. The coefficients in the Pieri rules are, except for an extra determinant, products of quotients of linear factors in $\alpha$ (expressed, as in the usual Jack polynomial case, in terms of certain hook-lengths in a Ferrers' diagram). We show that, surprisingly, the extra determinant is related to the partition function of the 6-vertex model. We give, as a conjecture, the Pieri rules for the Macdonald polynomials in superspace.

Skew Pieri algebras of the general linear group

Let V be an irreducible polynomial representation of the general linear group GLn=GLn(C) and let α1, …, αq be nonnegative integers less than or equal to n. We call a description of the irreducible decomposition of the tensor product V⊗Λα1(Cn)⊗⋯⊗Λαq(Cn) an iterated skew Pieri rule for GLn. In this paper, we define a family of complex algebras whose structure encodes an iterated skew Pieri rule for GLn, and we call these algebras iterated skew Pieri algebras. Our main goal is to construct a basis for each of these algebras thereby giving explicit highest weight vectors in the above tensor product.

Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs

Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. With this approach we prove skew Cauchy identity and then derive various applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also give a new explanation of the finite expansion property for products of Grothendieck polynomials.

An Orthosymplectic Pieri Rule

The classical Pieri formula gives a combinatorial rule for decomposing the product of a Schur function and a complete homogeneous symmetric polynomial as a linear combination of Schur functions with integer coefficients. We give a Pieri rule for describing the product of an orthosymplectic character and an orthosymplectic character arising from a one-row partition. We establish that the orthosymplectic Pieri rule coincides with Sundaram's Pieri rule for symplectic characters and that orthosymplectic characters and symplectic characters obey the same product rule.

Quasisymmetric and noncommutative skew Pieri rules

In this note we derive skew Pieri rules in the spirit of Assaf–McNamara for skew quasisymmetric Schur functions using the Hopf algebraic techniques of Lam–Lauve–Sottile, and recover the original rules of Assaf–McNamara as a special case. We then apply these techniques a second time to obtain skew Pieri rules for skew noncommutative Schur functions.

Structure Constants for Immaculate Functions

The immaculate functions, $\mathfrak{S}_{\alpha}$, were introduced as a Schur-like basis for $\operatorname{\mathsf{Nsym}}$. We investigate facts about their structure constants. These are analogues of Littlewood-Richardson coefficents. We will give a new proof of the left Pieri rule for the $\mathfrak{S}_{\alpha}$, a translation invariance property for the structure coefficients of the $\mathfrak{S}_{\alpha}$, and a counterexample to an $\mathfrak{S}_{\alpha}$-analogue of the saturation conjecture.

Contravariant form for reduction algebras

We study properties and constructions of contravariant forms on reduction algebras. As an application we compute norms of highest weight vectors in the tensor product of an irreducible finite dimensional representation of the Lie algebra gl(n) with a symmetric or wedge tensor power of its fundamental representation. Their zeroes describe Pieri rules.

Pieri integral formula and asymptotics of Jack unitary characters

We introduce Jack (unitary) characters and prove two kinds of formulas that are suitable for their asymptotics, as the lengths of the signatures that parametrize them go to infinity. The first kind includes several integral representations for Jack characters of one variable. The second identity we prove is the Pieri integral formula for Jack characters which, in a sense, is dual to the well known Pieri rule for Jack polynomials. The Pieri integral formula can also be seen as a functional equation for irreducible spherical functions of virtual Gelfand pairs. As an application of our formulas, we study the asymptotics of Jack characters as the corresponding signatures grow to infinity in the sense of Vershik–Kerov. We prove the existence of a small $$\delta > 0$$ such that the Jack characters of m variables have a uniform limit on the $$\delta $$ -neighborhood of the m-dimensional torus. Our result specializes to a theorem of Okounkov and Olshanski.

Shifted Hecke insertion and the K-theory of OG(n,2n + 1)

Patrias and Pylyavskyy introduced shifted Hecke insertion as an application of their theory of dual filtered graphs. We study shifted Hecke insertion, showing it preserves descent sets and relating it the K-theoretic jeu de taquin of Buch–Samuel and Clifford–Thomas–Yong. As a consequence, we construct symmetric functions that are closely related to Ikeda–Naruse's representatives for the K-theory of the orthogonal Grassmannian. Exploiting this relationship and introducing a shifted K-theoretic Poirier–Reutenauer algebra, we derive a Littlewood–Richardson rule for the K-theory of the orthogonal Grassmannian equivalent to the rules of Clifford–Thomas–Yong and Buch–Samuel. Our methods are independent of the Buch–Ravikumar Pieri rule.

Lowest 𝔰𝔩(2)-types in 𝔰𝔩(𝔫)-representations

Fix n ≥ 3 n \geq 3 . Let s \mathfrak {s} be a principally embedded s l 2 \mathfrak {sl}_2 -subalgebra in s l n \mathfrak {sl}_n . A special case of results of the second author and Gregg Zuckerman implies that there exists a positive integer b ( n ) b(n) such that for any finite dimensional irreducible s l n \mathfrak {sl}_n -representation, V V , there exists an irreducible s \mathfrak {s} -representation embedding in V V with dimension at most b ( n ) b(n) . We prove that b ( n ) = n b(n)=n is the sharpest possible bound. We also address embeddings other than the principal one. The exposition involves an application of the Cartan–Helgason theorem, Pieri rules, Hermite reciprocity, and a calculation in the “branching algebra” introduced by Roger Howe, Eng-Chye Tan, and the second author.

A reciprocity law and the skew Pieri rule for the symplectic group

We use the theory of skew duality to show that decomposing the tensor product of k irreducible representations of the symplectic group Sp2m=Sp2m(ℂ) is equivalent to branching from Sp2n to Sp2n1×⋯×Sp2nk, where n,n1,…,nk are positive integers such that n=n1+⋯+nk and the njs depend on m as well as the representations in the tensor product. Using this result and a work of Lepowsky, we obtain a skew Pieri rule for Sp2m, i.e., a description of the irreducible decomposition of the tensor product of an irreducible representation of the symplectic group Sp2m with a fundamental representation.

Pieri rules for Schur functions in superspace

The Schur functions in superspace sΛ and s¯Λ are the limits q=t=0 and q=t=∞ respectively of the Macdonald polynomials in superspace. We prove Pieri rules for the bases sΛ and s¯Λ (which happen to be essentially dual). As a consequence, we derive the basic properties of these bases such as dualities, monomial expansions, and tableaux generating functions.

Pieri Rules for Classical Groups and Equinumeration between Generalized Oscillating Tableaux and Semistandard Tableaux

We present several equinumerous results between generalized oscillating tableaux and semistandard tableaux and give a representation-theoretic proof to them. As one of the key ingredients of the proof, we provide Pieri rules for the symplectic and orthogonal groups.

Pieri and Littlewood–Richardson rules for two rows and cluster algebra structure

We study an algebra encoding a twice-iterated Pieri rule for the representations of the general linear group and prove that it has the structure of a cluster algebra. We also show that its cluster variables invariant under a unipotent subgroup generate the highest weight vectors of irreducible representations occurring in the decomposition of the tensor product of two irreducible representations of the general linear group one of whom is labeled by a Young diagram with less than or equal to two rows.

Pieri rule for the affine flag variety

We prove the affine Pieri rule for the cohomology of the affine flag variety conjectured by Lam, Lapointe, Morse and Shimozono. We study the cap operator on the affine nilHecke ring that is motivated by Kostant and Kumar's work on the equivariant cohomology of the affine flag variety. We show that the cap operators for Pieri elements are the same as Pieri operators defined by Berg, Saliola and Serrano. This establishes the affine Pieri rule. We also discuss properties of cap operators which are not necessarily affine A type.

A Murnaghan–Nakayama rule for noncommutative Schur functions

We prove a Murnaghan–Nakayama rule for noncommutative Schur functions introduced by Bessenrodt, Luoto and van Willigenburg. In other words, we give an explicit combinatorial formula for expanding the product of a noncommutative power sum symmetric function and a noncommutative Schur function in terms of noncommutative Schur functions. In direct analogy to the classical Murnaghan–Nakayama rule, the summands are computed using a noncommutative analogue of border strips, and have coefficients ±1 determined by the height of these border strips. The rule is proved by interpreting the noncommutative Pieri rules for noncommutative Schur functions in terms of box-adding operators on compositions.

Abacus-tournament models for Hall–Littlewood polynomials

In 2010, the first author introduced a combinatorial model for Schur polynomials based on labeled abaci. We generalize this construction to give analogous models for the Hall–Littlewood symmetric polynomials Pλ, Qλ, and Rλ using objects called abacus-tournaments. We introduce various cancellation mechanisms on abacus-tournaments to obtain simpler combinatorial formulas and explain why these polynomials are divisible by certain products of t-factorials. These tools are then applied to give bijective proofs of several identities involving Hall–Littlewood polynomials, including the Pieri rule that expands the product Pμek into a linear combination of Hall–Littlewood polynomials.