scholarly journals Character Polynomials, their q-Analogs and the Kronecker Product

10.37236/85 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
A. M. Garsia ◽  
A. Goupil
Keyword(s):  
Numerical Calculation ◽  
Kronecker Product ◽  
Hecke Algebra ◽  
Kronecker Products ◽  
Proper Role ◽  
Special Cases ◽  
Natural Counterpart ◽  
Kronecker Coefficients ◽  
Combinatorial Construction

The numerical calculation of character values as well as Kronecker coefficients can efficently be carried out by means of character polynomials. Yet these polynomials do not seem to have been given a proper role in present day literature or software. To show their remarkable simplicity we give here an "umbral" version and a recursive combinatorial construction. We also show that these polynomials have a natural counterpart in the standard Hecke algebra ${\cal H}_n(q\, )$. Their relation to Kronecker products is brought to the fore, as well as special cases and applications. This paper may also be used as a tutorial for working with character polynomials in the computation of Kronecker coefficients.

scholarly journals Stability of Kronecker Products of Irreducible Characters of the Symmetric Group

10.37236/1471 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Ernesto Vallejo
Keyword(s):  
Lower Bound ◽  
Symmetric Group ◽  
Kronecker Product ◽  
Kronecker Products ◽  
Irreducible Characters ◽  
Long Time ◽  
The Stability

F. Murnaghan observed a long time ago that the computation of the decompositon of the Kronecker product $\chi^{(n-a, \lambda_2, \dots )}\otimes \chi^{(n-b, \mu_2, \dots)}$ of two irreducible characters of the symmetric group into irreducibles depends only on $\overline\lambda=(\lambda_2,\dots )$ and $\overline\mu =(\mu_2,\dots )$, but not on $n$. In this note we prove a similar result: given three partitions $\lambda$, $\mu$, $\nu$ of $n$ we obtain a lower bound on $n$, depending on $\overline\lambda$, $\overline\mu$, $\overline\nu$, for the stability of the multiplicity $c(\lambda,\mu,\nu)$ of $\chi^\nu$ in $\chi^\lambda \otimes \chi^\mu$. Our proof is purely combinatorial. It uses a description of the $c(\lambda,\mu,\nu)$'s in terms of signed special rim hook tabloids and Littlewood-Richardson multitableaux.

scholarly journals The Computational Complexity of Plethysm Coefficients

2020 ◽  
Vol 29 (2) ◽  
Author(s):  
Nick Fischer ◽  
Christian Ikenmeyer
Keyword(s):  
Matrix Multiplication ◽  
Complexity Classes ◽  
Algebraic Complexity ◽  
Test Bed ◽  
Discrete Tomography ◽  
Geometric Complexity Theory ◽  
Special Cases ◽  
The Matrix ◽  
Kronecker Coefficients ◽  
Coordinate Rings

AbstractIn two papers, Bürgisser and Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix multiplication tensor. A key ingredient was information about certain Kronecker coefficients. While tensors are an interesting test bed for GCT ideas, the far-away goal is the separation of algebraic complexity classes. The role of the Kronecker coefficients in that setting is taken by the so-called plethysm coefficients: These are the multiplicities in the coordinate rings of spaces of polynomials. Even though several hardness results for Kronecker coefficients are known, there are almost no results about the complexity of computing the plethysm coefficients or even deciding their positivity.In this paper, we show that deciding positivity of plethysm coefficients is -hard and that computing plethysm coefficients is #-hard. In fact, both problems remain hard even if the inner parameter of the plethysm coefficient is fixed. In this way, we obtain an inner versus outer contrast: If the outer parameter of the plethysm coefficient is fixed, then the plethysm coefficient can be computed in polynomial time. Moreover, we derive new lower and upper bounds and in special cases even combinatorial descriptions for plethysm coefficients, which we consider to be of independent interest. Our technique uses discrete tomography in a more refined way than the recent work on Kronecker coefficients by Ikenmeyer, Mulmuley, and Walter (Comput Compl 2017). This makes our work the first to apply techniques from discrete tomography to the study of plethysm coefficients. Quite surprisingly, that interpretation also leads to new equalities between certain plethysm coefficients and Kronecker coefficients.

The correlated random walk with boundaries: A combinatorial solution

2000 ◽  
Vol 37 (2) ◽  
pp. 470-479 ◽  
Author(s):  
W. Böhm
Keyword(s):  
Random Walk ◽  
Asymptotic Formula ◽  
Lattice Paths ◽  
Correlated Random Walk ◽  
Absorbing Boundaries ◽  
Transition Functions ◽  
Special Cases ◽  
Combinatorial Construction

The transition functions for the correlated random walk with two absorbing boundaries are derived by means of a combinatorial construction which is based on Krattenthaler's theorem for counting lattice paths with turns. Results for walks with one boundary and for unrestricted walks are presented as special cases. Finally we give an asymptotic formula, which proves to be useful for computational purposes.

A proof of the linear Arithmetic Fundamental Lemma for

2020 ◽  
pp. 1-47
Author(s):  
Qirui Li
Keyword(s):  
Local Field ◽  
Intersection Number ◽  
Hecke Algebra ◽  
Quadratic Extension ◽  
Unit Element ◽  
Linear Arithmetic ◽  
Fundamental Lemma ◽  
Special Cases

Abstract Let $K/F$ be an unramified quadratic extension of a non-Archimedean local field. In a previous work [1], we proved a formula for the intersection number on Lubin–Tate spaces. The main result of this article is an algorithm for computation of this formula in certain special cases. As an application, we prove the linear Arithmetic Fundamental Lemma for $ \operatorname {{\mathrm {GL}}}_4$ with the unit element in the spherical Hecke Algebra.

scholarly journals Multidimensional Arrays, Indices and Kronecker Products

Econometrics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 18
Author(s):  
D. Stephen G. Pollock
Keyword(s):  
Matrix Algebra ◽  
Kronecker Product ◽  
Kronecker Products ◽  
Multidimensional Arrays ◽  
Conventional Notation ◽  
Index Notation ◽  
Product Of Matrices

Much of the algebra that is associated with the Kronecker product of matrices has been rendered in the conventional notation of matrix algebra, which conceals the essential structures of the objects of the analysis. This makes it difficult to establish even the most salient of the results. The problems can be greatly alleviated by adopting an orderly index notation that reveals these structures. This claim is demonstrated by considering a problem that several authors have already addressed without producing a widely accepted solution.

scholarly journals Combinatorial Gelfand Models

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Ron M. Adin ◽  
Alex Postnikov ◽  
Yuval Roichman
Keyword(s):  
Symmetric Group ◽  
Hecke Algebra ◽  
Hyperoctahedral Group ◽  
International Audience ◽  
Combinatorial Construction

International audience A combinatorial construction of Gelfand models for the symmetric group, for its Iwahori-Hecke algebra and for the hyperoctahedral group is presented.

scholarly journals Evaluations of Hecke Algebra Traces at Kazhdan-Lusztig Basis Elements

10.37236/5021 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Samuel Clearman ◽  
Matthew Hyatt ◽  
Brittany Shelton ◽  
Mark Skandera
Keyword(s):  
Hecke Algebra ◽  
Combinatorial Interpretation ◽  
Quasisymmetric Functions ◽  
Irreducible Characters ◽  
Algebraic Interpretation ◽  
Special Cases

For irreducible characters $\{ \chi_q^\lambda \,|\, \lambda \vdash n \}$, induced sign characters $\{ \epsilon_q^\lambda \,|\, \lambda \vdash n \}$, and induced trivial characters $\{ \eta_q^\lambda \,|\, \lambda \vdash n \}$ of the Hecke algebra $H_n(q)$, and Kazhdan-Lusztig basis elements $C'_w(q)$ with $w$ avoiding the patterns 3412 and 4231, we combinatorially interpret the polynomials $\smash{\chi_q^\lambda(q^{\ell(w)/2} C'_w(q))}$, $\smash{\epsilon_q^\lambda(q^{\ell(w)/2} C'_w(q))}$, and $\smash{\eta_q^\lambda(q^{\ell(w)/2} C'_w(q))}$. This gives a new algebraic interpretation of chromatic quasisymmetric functions of Shareshian and Wachs, and a new combinatorial interpretation of special cases of results of Haiman. We prove similar results for other $H_n(q)$-traces, and confirm a formula conjectured by Haiman.

Face Recognition Using 2D-FLDA Based on Approximate SVD Obtained with Kronecker Products with One Training Sample

2019 ◽  
pp. 1-9
Author(s):  
Ümit Çiğdem Turhal
Keyword(s):  
Face Recognition ◽  
Kronecker Product ◽  
Dimensional Space ◽  
Recognition Performance ◽  
Recognition Task ◽  
Linear Structure ◽  
Training Sample ◽  
Kronecker Products ◽  
Linear Discriminant ◽  
Lower Dimensional

Aims: In a face recognition task, it is a challenging problem to find lots of images for a person. Even, sometimes there can be only one image, available for a person. In these cases many of the methods are exposed to serious performance drops even some of these fail to work. Recently this problem has become remarkable for researchers. In some of these studies the database is extended using a synthesized image which is constructed from the singular value decomposition (SVD) of the single training image. In this paper, for such a method, SVD based 2 Dimensional Fisher Linear Discriminant Analysis (2D-FLDA), it is proposed a new approach to find the SVD of the image matrix with the aim of to increase the recognition performance. Study Design: In this paper, in a face recognition task with 2D-FLDA, in one training sample case, instead of original SVD of the image matrix, the approximate SVD of its based on multiple kronecker product sums is used. In order to obtain it, image matrix is first reshaped thus it is to be lower dimensional matrices and, then the sum of multiple kronecker products (MKPS) is applied in this lower dimensional space. Methodology: Experiments are performed on two known databases Ar-Face and ORL face databases. The performance of the proposed method is evaluated when there are facial expression, lightning conditions and pose variations. Results: In each experiment, the approximate SVD approach based on multiple kronecker product sum gets approximately 3% better results when compared with the original SVD. Conclusion: Experimental results verify that the proposed method achieves better recognition performance over the traditional one. The reason for this is the proposed approximate SVD has the advantages of simplicity, and also as the kronecker factors possess additional linear structure, kronecker product can capture potential self-similarity.

scholarly journals On the Kronecker Products and Their Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Huamin Zhang ◽  
Feng Ding
Keyword(s):  
Kronecker Product ◽  
Polynomial Matrix ◽  
Singular Values ◽  
Permutation Matrix ◽  
Kronecker Products ◽  
Mixed Matrix ◽  
Matrix Products ◽  
Vector Operator

This paper studies the properties of the Kronecker product related to the mixed matrix products, the vector operator, and the vec-permutation matrix and gives several theorems and their proofs. In addition, we establish the relations between the singular values of two matrices and their Kronecker product and the relations between the determinant, the trace, the rank, and the polynomial matrix of the Kronecker products.

Kronecker Products and Local Joins of Graphs

1977 ◽  
Vol 29 (2) ◽  
pp. 255-269 ◽  
Author(s):  
M. Farzan ◽  
D. A. Waller
Keyword(s):  
Tensor Product ◽  
Kronecker Product ◽  
Kronecker Products ◽  
Finite Graphs ◽  
Product Of Graphs

When studying the category raph of finite graphs and their morphisms, Ave can exploit the fact that this category has products, [we define these ideas in detail in § 2]. This categorical product of graphs is usually called their Kronecker product, though it has been approached by various authors in various ways and under various names, including tensor product, cardinal product, conjunction and of course categorical product (see for example [6; 7; 11 ; 14; 17 and 23]).

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