Pieri Rule Research Articles

Bernstein operators for universal characters and symplectic universal characters

This paper focuses on the construction of the Bernstein operators for universal characters and symplectic universal characters. By carrying out the action of a series of Bernstein operators on the constant function 1, universal characters and symplectic universal characters have been presented. Furthermore, we concentrate on the investigation of the product of universal characters (symplectic universal characters) and a complete symmetric function. Based upon the Pieri rules and inverse Pieri rules, universal characters and symplectic universal characters can be derived by changing corresponding diagrams of partitions.

Toward the quantum Pieri rule for

Toward the quantum Pieri rule for

Pieri rules for skew dual immaculate functions

Abstract In this paper, we give Pieri rules for skew dual immaculate functions and their recently discovered row-strict counterparts. We establish our rules using a right-action analogue of the skew Littlewood–Richardson rule for Hopf algebras of Lam–Lauve–Sottile. We also obtain Pieri rules for row-strict (dual) immaculate functions.

Symmetry and Pieri rules for the bisymmetric Macdonald polynomials

Bisymmetric Macdonald polynomials can be obtained through a process of antisymmetrization and t-symmetrization of non-symmetric Macdonald polynomials. Using the double affine Hecke algebra, we show that the evaluation of the bisymmetric Macdonald polynomials satisfies a symmetry property generalizing that satisfied by the usual Macdonald polynomials. We then obtain Pieri rules for the bisymmetric Macdonald polynomials where the sums are over certain vertical strips.

Super-Schur polynomials for Affine Super Yangian Y( hat{mathfrak{gl}} 1|1)

We explicitly construct cut-and-join operators and their eigenfunctions — the Super-Schur functions — for the case of the affine super-Yangian Y( hat{mathfrak{gl}} 1|1). This is the simplest non-trivial (semi-Fock) representation, where eigenfunctions are labeled by the superanalogue of 2d Young diagrams, and depend on the supertime variables (pk, θk). The action of other generators on diagrams is described by the analogue of the Pieri rule. As well we present generalizations of the hook formula for the measure on super-Young diagrams and of the Cauchy formula. Also a discussion of string theory origins for these relations is provided.

A Generalization of the Murnaghan–Nakayama Rule for K-k-Schur and k-Schur Functions

Abstract The $K$-$k$-Schur functions and $k$-Schur functions appeared in the study of $K$-theoretic and affine Schubert Calculus as polynomial representatives of Schubert classes. In this paper, we introduce a new family of symmetric functions $\mathcal {F}_{\lambda }^{(k)}$, that generalizes the constructions via the Pieri rule of $K$-$k$-Schur functions and $ k$-Schur functions. Then we obtain the Murnaghan–Nakayama rule for the generalized functions. The rule is described explicitly in the cases of $K$-$k$-Schur functions and $k$-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for $k$-Schur functions, and explains it as a degeneration of the rule for $K$-$k$-Schur functions. In particular, many other special cases and connections promise to be detailed in the future.

Murnaghan–Nakayama Rule and Spin Bitrace for the Hecke–Clifford Algebra

AbstractA Pfaffian-type Murnaghan–Nakayama rule is derived for the Hecke–Clifford algebra $\mathcal{H}^{c}_{n}$ based on the Frobenius formula and vertex operators, and this leads to a combinatorial version via the tableaux realization of Schur’s $Q$-functions. As a consequence, a general formula for the irreducible characters $\zeta ^{\lambda }_{\mu }(q)$ using partition-valued functions is derived. Meanwhile, an iterative formula on the indexing partition $\lambda $ via the Pieri rule is also deduced. As applications, some compact formulae of the irreducible characters are given for special partitions and a symmetric property of the irreducible character is found. We also introduce the spin bitrace as the analogue of the bitrace for the Hecke algebra and derive its general combinatorial formula. Tables of irreducible characters are listed for $n\leq 7.$

Twisted Schubert polynomials

We prove that twisted versions of Schubert polynomials defined by $$\widetilde{\mathfrak S}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}$$ and $$\widetilde{\mathfrak S}_{ws_i} = (s_i+\partial _i)\widetilde{\mathfrak S}_w$$ are monomial positive and give a combinatorial formula for their coefficients. In doing so, we reprove and extend a previous result about positivity of skew divided difference operators and show how it implies the Pieri rule for Schubert polynomials. We also give positive formulas for double versions of the $$\widetilde{\mathfrak S}_w$$ as well as their localizations.

Semifinite Harmonic Functions on the Zigzag Graph

We study semifinite harmonic functions on the zigzag graph, which corresponds to the Pieri rule for the fundamental quasisymmetric functions $$\{F_{\lambda}\}$$ . The main problem, which we solve here, is to classify the indecomposable semifinite harmonic functions on this graph. We show that these functions are in a natural bijective correspondence with some combinatorial data, the so-called semifinite zigzag growth models. Furthermore, we describe an explicit construction that produces a semifinite indecomposable harmonic function from every semifinite zigzag growth model. We also establish a semifinite analogue of the Vershik–Kerov ring theorem.

An Equivariant Quantum Pieri Rule for the Grassmannian on Cylindric Shapes

The quantum cohomology ring of the Grassmannian is determined by the quantum Pieri rule for multiplying by Schubert classes indexed by row or column-shaped partitions. We provide a direct equivariant generalization of Postnikov's quantum Pieri rule for the Grassmannian in terms of cylindric shapes, complementing related work of Gorbounov and Korff in quantum integrable systems. The equivariant terms in our Graham-positive rule simply encode the positions of all possible addable boxes within one cylindric skew diagram. As such, unlike the earlier equivariant quantum Pieri rule of Huang and Li and known equivariant quantum Littlewood-Richardson rules, our formula does not require any calculations in a different Grassmannian or two-step flag variety.

Some algebraic structures in KPZ universality

We review some algebraic and combinatorial structures that underlie models in the KPZ universality class. Emphasis is placed on the Robinson-Schensted-Knuth correspondence and its geometric lifting due to A.N.Kirillov. We present how these combinatorial constructions are used to analyse the structure of solvable models in the KPZ class and lead to computation of their statistics via connecting to representation theoretic objects such as Schur, Macdonald and Whittaker functions, Young tableaux and Gelfand-Tsetlin patterns. We also present how fundamental representation theoretic concepts, such as the Cauchy identity, the Pieri rule and the branching rule, can be used, alongside RSK correspondences, and can be combined with probabilistic ideas, in order to construct integrable stochastic dynamics on two dimensional arrays of Gelfand-Tsetlin type, in ways that couple different one dimensional stochastic processes. For example, interacting particle systems, on the one hand, and processes related to eigenvalues of random matrices, on the other, thus illuminating the emergence of random matrix distributions in interacting stochastic processes. The goal of the notes is to expose some of the overarching principles, which have driven a significant number of developments in the field.

Affine Pieri rule for periodic Macdonald spherical functions and fusion rings

Let gˆ be an untwisted affine Lie algebra or the twisted counterpart thereof (which excludes the affine Lie algebras of type BCˆn=A2n(2)). We present an affine Pieri rule for a basis of periodic Macdonald spherical functions associated with gˆ. In type Aˆn−1=An−1(1) the formula in question reproduces an affine Pieri rule for cylindric Hall-Littlewood polynomials due to Korff, which at t=0 specializes in turn to a well-known Pieri formula in the fusion ring of genus zero slˆ(n)c-Wess-Zumino-Witten conformal field theories.

Explicit Pieri Inclusions

By the Pieri rule, the tensor product of an exterior power and a finite-dimensional irreducible representation of a general linear group has a multiplicity-free decomposition. The embeddings of the constituents are called Pieri inclusions and were first studied by Weyman in his thesis and described explicitly by Olver. More recently, these maps have appeared in the work of Eisenbud, Fløystad, and Weyman and of Sam and Weyman to compute pure free resolutions for classical groups.
 In this paper, we give a new closed form, non-recursive description of Pieri inclusions. For partitions with a bounded number of distinct parts, the resulting algorithm has polynomial time complexity whereas the previously known algorithm has exponential time complexity.

Spin q–Whittaker polynomials

We introduce and study a one-parameter generalization of the q–Whittaker symmetric functions. This is a family of multivariate symmetric polynomials, whose construction may be viewed as an application of the procedure of fusion from integrable lattice models to a vertex model interpretation of a one-parameter generalization of Hall–Littlewood polynomials from [3,6,7].We prove branching and Pieri rules, standard and dual (skew) Cauchy summation identities, and an integral representation for the new polynomials.

Honeycombs for Hall Polynomials

We propose a new formulation of Hall polynomials in terms of honeycombs, which were previously introduced in the context of the Littlewood–Richardson rule. We prove a Pieri rule and associativity for our honeycomb formula, thus showing equality with Hall polynomials. Our proofs are linear algebraic in nature, extending nontrivially the corresponding bijective results for ordinary Littlewood–Richardson coefficients [A. Knutson, T. Tao, C. Woodward, 2004].

Kraskiewicz-Pragacz modules and Pieri and dual Pieri rules for Schubert polynomials

In their 1987 paper Kraskiewicz and Pragacz defined certain modules, which we call KP modules, over the upper triangular Lie algebra whose characters are Schubert polynomials. In a previous work the author showed that the tensor product of Kraskiewicz-Pragacz modules always has KP filtration, i.e. a filtration whose each successive quotients are isomorphic to KP modules. In this paper we explicitly construct such filtrations for certain special cases of these tensor product modules, namely Sw Sd(Ki) and Sw Vd(Ki), corresponding to Pieri and dual Pieri rules for Schubert polynomials.

Symmetric elliptic functions, IRF models, and dynamic exclusion processes

We introduce stochastic Interaction-Round-a-Face (IRF) models that are related to representations of the elliptic quantum group E_{\tau, \eta}(sl_2) . For stochastic IRF models in a quadrant, we evaluate averages for a broad family of observables that can be viewed as higher analogs of q -moments of the height function for the stochastic (higher spin) six vertex models. In a certain limit, the stochastic IRF models degenerate to (1+1)d interacting particle systems that we call dynamic ASEP and SSEP; their jump rates depend on local values of the height function. For the step initial condition, we evaluate averages of observables for them as well, and use those to investigate one-point asymptotics of the dynamic SSEP. The construction and proofs are based on remarkable properties (branching and Pieri rules, Cauchy identities) of a (seemingly new) family of symmetric elliptic functions that arise as matrix elements in an infinite volume limit of the algebraic Bethe ansatz for E_{\tau, \eta}(sl_2) .

Dual graphs from noncommutative and quasisymmetric Schur functions

By establishing relations between operators on compositions, we show that the posets of compositions arising from the right and left Pieri rules for noncommutative Schur functions can each be endowed with both the structure of dual graded graphs and dual filtered graphs when paired with the poset of compositions arising from the Pieri rules for quasisymmetric Schur functions and its deformation.

Rigidity for the Hopf Algebra of Quasisymmetric Functions

We investigate the rigidity for the Hopf algebra QSym of quasisymmetric functions with respect to the monomial, the fundamental and the quasisymmetric Schur basis, respectively. By establishing some combinatorial properties of the posets of compositions arising from the analogous Pieri rules for quasisymmetric functions, we show that QSym is rigid as an algebra with respect to the quasisymmetric Schur basis, and rigid as a coalgebra with respect to the monomial and the quasisymmetric Schur basis, respectively. The natural actions of reversal, complement and transpose of the labelling compositions lead to some nontrivial graded (co)algebra automorphisms of QSym. We prove that the linear maps induced by the three actions are precisely the only nontrivial graded algebra automorphisms that take the fundamental basis into itself. Furthermore, the complement map on the labels gives the unique nontrivial graded coalgebra automorphism preserving the fundamental basis, while the reversal map on the labels gives the unique nontrivial graded algebra automorphism preserving the monomial basis. Therefore, QSym is rigid as a Hopf algebra with respect to the monomial and the quasisymmetric Schur basis.

The Boundary of the Refined Kingman Graph

We study the refined Kingman graph 𝔻, first introduced by Gnedin, whose vertices are indexed by the set of compositions of positive integers and multiplicity function reflects the Pieri rule for quasisymmetric monomial functions. Gnedin identified the Martin boundary of 𝔻 with the space Ω of sets of disjoint open subintervals of [0, 1]. We show that the minimal and Martin boundaries of 𝔻 coincide.