scholarly journals Some Singular Vector-Valued Jack and Macdonald Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 503
Author(s):  
Charles Dunkl
Keyword(s):  
Symmetric Groups ◽  
Infinite Family ◽  
Hecke Algebra ◽  
Singular Values ◽  
Macdonald Polynomials ◽  
Complex Reflection Groups ◽  
Leading Term ◽  
Standard Young Tableaux ◽  
Standard Young Tableau ◽  
Associated Spaces

For each partition τ of N, there are irreducible modules of the symmetric groups S N and of the corresponding Hecke algebra H N t whose bases consist of the reverse standard Young tableaux of shape τ . There are associated spaces of nonsymmetric Jack and Macdonald polynomials taking values in these modules. The Jack polynomials form a special case of the polynomials constructed by Griffeth for the infinite family G n , p , N of complex reflection groups. The Macdonald polynomials were constructed by Luque and the author. For each of the groups S N and the Hecke algebra H N t , there is a commutative set of Dunkl operators. The Jack and the Macdonald polynomials are parametrized by κ and q , t , respectively. For certain values of these parameters (called singular values), there are polynomials annihilated by each Dunkl operator; these are called singular polynomials. This paper analyzes the singular polynomials whose leading term is x 1 m ⊗ S , where S is an arbitrary reverse standard Young tableau of shape τ . The singular values depend on the properties of the edge of the Ferrers diagram of τ .

scholarly journals Nonsymmetric Macdonald Superpolynomials

2021 ◽  
Author(s):  
Charles F. Dunkl ◽  
Keyword(s):  
Classical Group ◽  
Young Tableau ◽  
Hecke Algebra ◽  
Type A ◽  
Macdonald Polynomials ◽  
Symmetric Polynomial ◽  
Inner Product ◽  
Symmetric Polynomials ◽  
Special Points ◽  
Standard Young Tableau

There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald polynomials taking values in arbitrary modules of the Hecke algebra. In this paper the two ideas are combined to define and study nonsymmetric Macdonald polynomials taking values in the aforementioned anti-commuting polynomials, in other words, superpolynomials. The modules, their orthogonal bases and their properties are first derived. In terms of the standard Young tableau approach to representations these modules correspond to hook tableaux. The details of the Dunkl-Luque theory and the particular application are presented. There is an inner product on the polynomials for which the Macdonald polynomials are mutually orthogonal. The squared norms for this product are determined. By using techniques of Baker and Forrester [Ann. Comb. 3 (1999), 159-170, arXiv:q-alg/9707001] symmetric Macdonald polynomials are built up from the nonsymmetric theory. Here ''symmetric'' means in the Hecke algebra sense, not in the classical group sense. There is a concise formula for the squared norm of the minimal symmetric polynomial, and some formulas for anti-symmetric polynomials. For both symmetric and anti-symmetric polynomials there is a factorization when the polynomials are evaluated at special points.

scholarly journals Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 779
Author(s):  
Charles F. Dunkl
Keyword(s):  
Preceding Paper ◽  
Hecke Algebra ◽  
Type A ◽  
Macdonald Polynomials ◽  
Special Points ◽  
Two Parameters

In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type A (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anti-commuting variables, to define nonsymmetric Macdonald superpolynomials. These polynomials depend on two parameters q,t and are defined by means of a Yang–Baxter graph. The present paper determines the values of a subclass of the polynomials at the special points 1,t,t2,… or 1,t−1,t−2,…. The arguments use induction on the degree and computations with products of generators of the Hecke algebra. The resulting formulas involve q,t-hook products. Evaluations are also found for Macdonald superpolynomials having restricted symmetry and antisymmetry properties.

scholarly journals Computing mu-values for Representations of Symmetric Groups in Engineering Systems

2018 ◽  
Vol 11 (3) ◽  
pp. 774-792
Author(s):  
Mutti-Ur Rehman ◽  
M. Fazeel Anwar
Keyword(s):  
Control Systems ◽  
Symmetric Groups ◽  
Singular Values ◽  
Linear Control ◽  
Linear Control Systems ◽  
Complex Numbers ◽  
Engineering Systems ◽  
Matrix Representations ◽  
The Matrix

In this article we consider the matrix representations of finite symmetric groups Sn over the filed of complex numbers. These groups and their representations also appear as symmetries of certain linear control systems [5]. We compute the structure singular values (SSV) of the matrices arising from these representations. The obtained results of SSV are compared with well-known MATLAB routine mussv.

scholarly journals Enumeration of Standard Young Tableaux of Shifted Strips with Constant Width

10.37236/6466 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Ping Sun
Keyword(s):  
Integral Method ◽  
Recurrence Relations ◽  
Young Tableau ◽  
Constant Width ◽  
Young Tableaux ◽  
Pell Number ◽  
Standard Young Tableaux ◽  
Standard Young Tableau

Let $g_{n_1,n_2}$ be the number of standard Young tableau of truncated shifted shape with $n_1$ rows and $n_2$ boxes in each row. By using the integral method this paper derives the recurrence relations of $g_{3,n}$, $g_{n,4}$ and $g_{n,5}$ respectively. Specifically, $g_{n,4}$ is the $(2n-1)$-st Pell number.

scholarly journals Murnaghan-Nakayama Rules for Characters of Iwahori-Hecke Algebras of the Complex Reflection Groups G(r, p, n)

1998 ◽  
Vol 50 (1) ◽  
pp. 167-192 ◽  
Author(s):  
Tom Halverson ◽  
Arun Ram
Keyword(s):  
Symmetric Group ◽  
Infinite Series ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Diagonal Matrix ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Irreducible Characters ◽  
Complex Reflection ◽  
Complex Reflection Groups

AbstractIwahori-Hecke algebras for the infinite series of complex reflection groups G(r, p, n) were constructed recently in the work of Ariki and Koike [AK], Broué andMalle [BM], and Ariki [Ari]. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper [HR] in whichwe derivedMurnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene [Gre], who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of G(r, p, n) given by Ariki and Koike [AK] and Ariki [Ari].

scholarly journals The Calogero-Moser partition for G(m, d, n)

2012 ◽  
Vol 207 ◽  
pp. 47-77 ◽  
Author(s):  
Gwyn Bellamy
Keyword(s):  
Hecke Algebra ◽  
Irreducible Representations ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Cyclotomic Hecke Algebra

AbstractWe show that it is possible to deduce the Calogero-Moser partition of the irreducible representations of the complex reflection groups G(m,d, n) from the corresponding partition for G(m,1,n). This confirms, in the case W = G(m,d,n), a conjecture of Gordon and Martino relating the Calogero-Moser partition to Rouquier families for the corresponding cyclotomic Hecke algebra.

scholarly journals Singular Nonsymmetric Jack Polynomials for Some Rectangular Tableaux

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 630
Author(s):  
Charles F. Dunkl
Keyword(s):  
Tensor Product ◽  
Symmetric Groups ◽  
Young Tableaux ◽  
Jack Polynomials ◽  
Main Ingredient ◽  
Dunkl Operators ◽  
Rational Cherednik Algebra ◽  
Standard Young Tableaux ◽  
Vector Valued

In the intersection of the theories of nonsymmetric Jack polynomials in N variables and representations of the symmetric groups S N one finds the singular polynomials. For certain values of the parameter κ there are Jack polynomials which span an irreducible S N -module and are annihilated by the Dunkl operators. The S N -module is labeled by a partition of N, called the isotype of the polynomials. In this paper the Jack polynomials are of the vector-valued type, i.e., elements of the tensor product of the scalar polynomials with the span of reverse standard Young tableaux of the shape of a fixed partition of N. In particular, this partition is of shape m , m , … , m with 2 k components and the constructed singular polynomials are of isotype m k , m k for the parameter κ = 1 / m + 2 . This paper contains the necessary background on nonsymmetric Jack polynomials and representation theory and explains the role of Jucys–Murphy elements in the construction. The main ingredient is the proof of uniqueness of certain spectral vectors, namely the list of eigenvalues of the Jack polynomials for the Cherednik–Dunkl operators, when specialized to κ = 1 / m + 2 . The paper finishes with a discussion of associated maps of modules of the rational Cherednik algebra and an example illustrating the difficulty of finding singular polynomials for arbitrary partitions.

scholarly journals On the Schur Expansion of Hall-Littlewood and Related Polynomials via Yamanouchi Words

10.37236/6732 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Austin Roberts
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Macdonald Polynomials ◽  
Macdonald Polynomial ◽  
Gamma Delta ◽  
Colored Graph ◽  
Littlewood Polynomials ◽  
Dual Equivalence ◽  
Leading Term ◽  
Definition Of

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $\delta \subset {\mathbb Z} \times {\mathbb Z}$, written as $\widetilde H_{\delta}(X;q,t)$ and $\widetilde H_{\delta}(X;0,t)$, respectively. We then give an explicit Schur expansion of $\widetilde H_{\delta}(X;0,t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function $R_{\gamma,\delta}(X)$ as a refinement of $\widetilde H_{\delta}(X;0,t)$ and similarly describe its Schur expansion. We then analyze $R_{\gamma,\delta}(X)$ to determine the leading term of its Schur expansion. We also provide a conjecture towards the Schur expansion of $\widetilde H_{\delta}(X;q,t)$. To gain these results, we use a construction from the 2007 work of Sami Assaf to associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_\delta$. In the case where a subgraph of $\mathcal{H}_\delta$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.

scholarly journals A Sundaram type Bijection for SO(3): Vacillating Tableaux and Pairs of Standard Young Tableaux and Orthogonal Littlewood-Richardson Tableaux

10.37236/7713 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Judith Jagenteufel
Keyword(s):  
Orthogonal Group ◽  
Young Tableau ◽  
Young Tableaux ◽  
Tensor Power ◽  
Direct Sum ◽  
Special Orthogonal Group ◽  
Direct Sum Decomposition ◽  
Descent Set ◽  
Standard Young Tableaux ◽  
Standard Young Tableau

Motivated by the direct-sum-decomposition of the $r^{\text{th}}$ tensor power of the defining representation of the special orthogonal group $\mathrm{SO}(2k + 1)$, we present a bijection between vacillating tableaux and pairs consisting of a standard Young tableau and an orthogonal Littlewood-Richardson tableau for $\mathrm{SO}(3)$.Our bijection preserves a suitably defined descent set. Using it we determine the quasi-symmetric expansion of the Frobenius characters of the isotypic components.On the combinatorial side we obtain a bijection between Riordan paths and standard Young tableaux with 3 rows, all of even length or all of odd length.

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