The q-deformation of symmetric functions and the symmetric group

The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric function. An explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing the chromatic quasisymmetric function of the incomparability graph of a natural unit interval order in terms of power sum symmetric functions, is proven. The proof uses a formula of Roichman for the irreducible characters of the symmetric group.

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Cohomology Classes of Interval Positroid Varieties and a Conjecture of Liu

The Electronic Journal of Combinatorics ◽

10.37236/6960 ◽

2018 ◽

Vol 25 (4) ◽

Author(s):

Brendan Pawlowski

Keyword(s):

Symmetric Group ◽

Finite Subset ◽

Upper Bound ◽

Cohomology Class ◽

Symmetric Functions ◽

A Priori ◽

Specht Module ◽

Group A ◽

Schubert Classes ◽

Stanley Symmetric Functions

To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ 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. To do this, we exhibit the appropriate diagram variety as a component in a degeneration of one of Knutson's interval positroid varieties (up to Grassmann duality). A priori, the cohomology classes of these interval positroid varieties are represented by affine Stanley symmetric functions. We give a different formula for these classes as ordinary Stanley symmetric functions, one with the advantage of being Schur-positive and compatible with inclusions between Grassmannians.

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Harmonic Functions on Multiplicative Graphs and Interpolation Polynomials

The Electronic Journal of Combinatorics ◽

10.37236/1506 ◽

2000 ◽

Vol 7 (1) ◽

Cited By ~ 21

Author(s):

Alexei Borodin ◽

Grigori Olshanski

Keyword(s):

Symmetric Group ◽

Harmonic Analysis ◽

Harmonic Functions ◽

Symmetric Functions ◽

Multivariate Interpolation ◽

Infinite Symmetric Group ◽

Interpolation Polynomials

We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur's S and P functions and with Jack symmetric functions. As a by–product, we compute certain Selberg–type integrals.

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Group characters and algebra

Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character ◽

10.1098/rsta.1934.0015 ◽

1934 ◽

Vol 233 (721-730) ◽

pp. 99-141 ◽

Cited By ~ 155

Keyword(s):

Symmetric Group ◽

Symmetric Functions ◽

Irreducible Representations ◽

Close Analogy ◽

Group Characters ◽

And Algebra

It has been known for some time* that the elements of a matrix of degree n may be arranged in sets which correspond to cycles of the symmetric group of order n !, and that there are relations connecting permanents and determinants, e.g. , /a* p y S a |3 y 8/ ( Further, MACMAHON and BRIOSCHI have pointed out the close analogy which exists between the threefold algebra of the symmetric functions an, hn and sn, and the theory of determinants, permanents, and the cycles of substitutions of the symmetric group. Here we trace the analogy to its source by fixing attention on the characters of the irreducible representations of the symmetric group of linear substitutions, as the centre of the whole theory. By this means divers theories of combinatory analysis and algebra are seen to be merely different aspects of the same theory. For the symmetric group of order n ! the characters are all integers, and we associate with each partition of n both a character of the group and a cycle of substitutions.

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On the square root of the centre of the Hecke algebra of type A

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700016037 ◽

2007 ◽

Vol 82 (2) ◽

pp. 209-220

Author(s):

Andrew Francis ◽

Lenny Jones

Keyword(s):

Symmetric Group ◽

Symmetric Functions ◽

Hecke Algebra ◽

Square Root ◽

Minimal Basis ◽

Elementary Symmetric Functions ◽

Square Roots ◽

Commutative Subalgebra ◽

Central Elements ◽

Generic Existence

AbstractIn this paper we investigate non-central elements of the Iwahori-Hecke algebra of the symmetric group whose squares are central. In particular, we describe a commutative subalgebra generated by certain non-central square roots of central elements, and the generic existence of a rank-three submodule of the Hecke algebra contained in the square root of the centre, but not in the centre. The generators for this commutative subalgebra include the longest word and elements related to trivial and sign characters of the Hecke algebra. We find elegant expressions for the squares of such generators in terms of both the minimal basis of the centre and the elementary symmetric functions of Murphy elements.

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A Characteristic Map for the Symmetric Space of Symplectic Forms Over a Finite Field

International Mathematics Research Notices ◽

10.1093/imrn/rnaa309 ◽

2020 ◽

Author(s):

Jimmy He

Keyword(s):

Symmetric Space ◽

Finite Field ◽

Symmetric Group ◽

Spherical Function ◽

Symmetric Functions ◽

Macdonald Polynomials ◽

Symplectic Forms ◽

Parabolic Induction ◽

Characteristic Map ◽

Combinatorial Formulas

Abstract The characteristic map for the symmetric group is an isomorphism relating the representation theory of the symmetric group to symmetric functions. An analogous isomorphism is constructed for the symmetric space of symplectic forms over a finite field, with the spherical functions being sent to Macdonald polynomials with parameters $(q,q^2)$. An analogue of parabolic induction is interpreted as a certain multiplication of symmetric functions. Applications are given to Schur positivity of skew Macdonald polynomials with parameters $(q,q^2)$ as well as combinatorial formulas for spherical function values.

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Generalized Pattern-Matching Conditions for

ISRN Combinatorics ◽

10.1155/2013/634823 ◽

2013 ◽

Vol 2013 ◽

pp. 1-20 ◽

Cited By ~ 1

Author(s):

Sergey Kitaev ◽

Andrew Niedermaier ◽

Jeffrey Remmel ◽

Manda Riehl

Keyword(s):

Symmetric Group ◽

Pattern Matching ◽

Infinite Number ◽

Generating Functions ◽

Symmetric Function ◽

Symmetric Functions ◽

Matching Condition ◽

Natural Analogues ◽

Generalized Pattern Matching ◽

Product Order

We derive several multivariable generating functions for a generalized pattern-matching condition on the wreath product of the cyclic group and the symmetric group . In particular, we derive the generating functions for the number of matches that occur in elements of for any pattern of length 2 by applying appropriate homomorphisms from the ring of symmetric functions over an infinite number of variables to simple symmetric function identities. This allows us to derive several natural analogues of the distribution of rises relative to the product order on elements of . Our research leads to connections to many known objects/structures yet to be explained combinatorially.

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