scholarly journals Branching rules in the ring of superclass functions of unipotent upper-triangular matrices

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Nathaniel Thiem
Keyword(s):  
Symmetric Functions ◽  
Combinatorial Structure ◽  
Classical Representation ◽  
Set Partitions ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Branching Rules ◽  
International Audience ◽  
Structure Constants ◽  
Supercharacter Theory

International audience It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group's relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters.

scholarly journals Nonzero coefficients in restrictions and tensor products of supercharacters of $U_n(q)$ (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Stephen Lewis ◽  
Nathaniel Thiem
Keyword(s):  
Tensor Product ◽  
Symmetric Group ◽  
Nonnegative Integer ◽  
Set Partitions ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Branching Rules ◽  
International Audience ◽  
Supercharacter Theory ◽  
Donne Lieu

International audience The standard supercharacter theory of the finite unipotent upper-triangular matrices $U_n(q)$ gives rise to a beautiful combinatorics based on set partitions. As with the representation theory of the symmetric group, embeddings of $U_m(q) \subseteq U_n(q)$ for $m \leq n$ lead to branching rules. Diaconis and Isaacs established that the restriction of a supercharacter of $U_n(q)$ is a nonnegative integer linear combination of supercharacters of $U_m(q)$ (in fact, it is polynomial in $q$). In a first step towards understanding the combinatorics of coefficients in the branching rules of the supercharacters of $U_n(q)$, this paper characterizes when a given coefficient is nonzero in the restriction of a supercharacter and the tensor product of two supercharacters. These conditions are given uniformly in terms of complete matchings in bipartite graphs. La théorie standard des supercaractères des matrices triangulaires supérieures unipotentes finies $U_n(q)$ donne lieu à une merveilleuse combinatoire basée sur les partitions d'ensembles. Comme avec la théorie des représentations du groupe symétrique, Les plongements $U_m(q) \subseteq U_n(q)$ pour $m \leq n$ mènent aux règles de branchement. Diaconis et Isaacs ont montré que la restriction d'un supercaractère de $U_n(q)$ est une combinaison linéaire des supercaractères de $U_m(q)$ avec des coefficients entiers non négatifs (en fait, elle est polynomiale en $q$). Dans une première étape vers la compréhension de la combinatoire des coefficients dans les règles de branchement des supercaractères de $U_n(q)$, ce texte caractérise les coefficients non nuls dans la restriction d'un supercaractère et dans le produit des tenseurs de deux supercaractères. Ces conditions sont données uniformément en termes des couplages complets dans des graphes bipartis.

scholarly journals Shell Tableaux: A Set Partition Analog of Vacillating Tableaux

10.37236/8241 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Megan Ly
Keyword(s):  
Irreducible Representations ◽  
Character Theory ◽  
Set Partition ◽  
Set Partitions ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Weyl Duality ◽  
Supercharacter Theory ◽  
Combinatorial Representation Theory ◽  
Centralizer Algebra

Schur–Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analog of Schur–Weyl duality for the group of unipotent upper triangular matrices over a finite field.  In this case, the character theory of these upper triangular matrices is "wild" or unattainable. Thus we employ a generalization, known as supercharacter theory, that creates a striking variation on the character theory of the symmetric group with combinatorics built from set partitions. In this paper, we present a combinatorial formula for calculating a restriction and induction of supercharacters based on statistics of set partitions and seashell inspired diagrams. We use these formulas to create a graph that encodes the decomposition of a tensor space, and develop an analog of Young tableaux, known as shell tableaux, to index paths in this graph. 

scholarly journals Restricting Supercharacters of the Finite Group of Unipotent Uppertriangular Matrices

10.37236/112 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Nathaniel Thiem ◽  
Vidya Venkateswaran
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Finite Field ◽  
Symmetric Group ◽  
Irreducible Representations ◽  
Combinatorial Theory ◽  
Set Partitions ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Supercharacter Theory

It is well-known that understanding the representation theory of the finite group of unipotent upper-triangular matrices $U_n$ over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one obtains a rich combinatorial theory analogous to that of the symmetric group, where we replace partition combinatorics with set-partitions. This paper studies the supercharacter theory of a family of subgroups that interpolate between $U_{n-1}$ and $U_n$. We supply several combinatorial indexing sets for the supercharacters, supercharacter formulas for these indexing sets, and a combinatorial rule for restricting supercharacters from one group to another. A consequence of this analysis is a Pieri-like restriction rule from $U_n$ to $U_{n-1}$ that can be described on set-partitions (analogous to the corresponding symmetric group rule on partitions).

scholarly journals Interval positroid varieties and a deformation of the ring of symmetric functions

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Allen Knutson ◽  
Mathias Lederer
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Dimensional Subspace ◽  
Schubert Varieties ◽  
International Audience ◽  
Structure Constants ◽  
Schubert Classes ◽  
K Theory ◽  
Two Parameter ◽  
Parameter Deformation

International audience Define the <b>interval rank</b> $r_[i,j] : Gr_k(\mathbb C^n) →\mathbb{N}$ of a k-plane V as the dimension of the orthogonal projection $π _[i,j](V)$ of V to the $(j-i+1)$-dimensional subspace that uses the coordinates $i,i+1,\ldots,j$. By measuring all these ranks, we define the <b>interval rank stratification</b> of the Grassmannian $Gr_k(\mathbb C^n)$. It is finer than the Schubert and Richardson stratifications, and coarser than the positroid stratification studied by Lusztig, Postnikov, and others, so we call the closures of these strata <b>interval positroid varieties</b>. We connect Vakil's "geometric Littlewood-Richardson rule", in which he computed the homology classes of Richardson varieties (Schubert varieties intersected with opposite Schubert varieties), to Erd&odblac;s-Ko-Rado shifting, and show that all of Vakil's varieties are interval positroid varieties. We build on his work in three ways: (1) we extend it to arbitrary interval positroid varieties, (2) we use it to compute in equivariant K-theory, not just homology, and (3) we simplify Vakil's (2+1)-dimensional "checker games" to 2-dimensional diagrams we call "IP pipe dreams". The ring Symm of symmetric functions and its basis of Schur functions is well-known to be very closely related to the ring $\bigoplus_a,b H_*(Gr_a(\mathbb{C}^{(a+b)})$ and its basis of Schubert classes. We extend the latter ring to equivariant K-theory (with respect to a circle action on each $\mathbb{C}^{(a+b)}$, and compute the structure constants of this two-parameter deformation of Symm using the interval positroid technology above.

scholarly journals Counting self-dual interval orders

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Vít Jelínek
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Interval Orders ◽  
Maximal Elements ◽  
Triangular Matrices ◽  
Bijective Correspondence ◽  
Upper Triangular Matrices ◽  
Positive Entry ◽  
International Audience ◽  
Natural Statistics

International audience In this paper, we first derive an explicit formula for the generating function that counts unlabeled interval orders (a.k.a. (2+2)-free posets) with respect to several natural statistics, including their size, magnitude, and the number of minimal and maximal elements. In the second part of the paper, we derive a generating function for the number of self-dual unlabeled interval orders, with respect to the same statistics. Our method is based on a bijective correspondence between interval orders and upper-triangular matrices in which each row and column has a positive entry. Dans cet article, on obtient une expression explicite pour la fonction génératrice du nombre des ensembles partiellement ordonnés (posets) qui évitent le motif (2+2). La fonction compte ces ensembles par rapport à plusieurs statistiques naturelles, incluant le nombre d'éléments, le nombre de niveaux, et le nombre d'éléments minimaux et maximaux. Dans la deuxième partie, on obtient une expression similaire pour la fonction génératrice des posets autoduaux évitant le motif (2+2). On obtient ces résultats à l'aide d'une bijection entre les posets évitant (2+2) et les matrices triangulaires supérieures dont chaque ligne et chaque colonne contient un élément positif.

scholarly journals A Plethysm Formula on the Characteristic Map of Induced Linear Characters from $U_n(\mathbb F_q)$ to $GL(\mathbb F_q)$

10.37236/2364 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Zhi Chen
Keyword(s):  
Finite Field ◽  
Recurrence Relation ◽  
Linear Group ◽  
General Linear Group ◽  
Symmetric Functions ◽  
General Linear ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Characteristic Map

This paper gives a plethysm formula on the characteristic map of the induced linear characters from the unipotent upper-triangular matrices $U_n(\mathbb F_q)$ to $GL_n(\mathbb F_q)$, the general linear group over finite field $\mathbb F_q$. The result turns out to be a multiple of a twisted version of the Hall-Littlewood symmetric functions $\tilde{P}_n[Y;q]$. A recurrence relation is also given which makes it easy to carry out the computation.

scholarly journals Ascent Sequences and Upper Triangular Matrices Containing Non-Negative Integers

10.37236/325 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Mark Dukes ◽  
Robert Parviainen
Keyword(s):  
Simple Condition ◽  
Set Partitions ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Classes Of Matrices ◽  
Zero Entry ◽  
Binary Matrices ◽  
Ascent Sequences ◽  
Natural Statistics ◽  
Special Classes

This paper presents a bijection between ascent sequences and upper triangular matrices whose non-negative entries are such that all rows and columns contain at least one non-zero entry. We show the equivalence of several natural statistics on these structures under this bijection and prove that some of these statistics are equidistributed. Several special classes of matrices are shown to have simple formulations in terms of ascent sequences. Binary matrices are shown to correspond to ascent sequences with no two adjacent entries the same. Bidiagonal matrices are shown to be related to order-consecutive set partitions and a simple condition on the ascent sequences generate this class.

scholarly journals A SUPERCHARACTER TABLE DECOMPOSITION VIA POWER-SUM SYMMETRIC FUNCTIONS

2013 ◽  
Vol 23 (04) ◽  
pp. 763-778 ◽  
Author(s):  
NANTEL BERGERON ◽  
NATHANIEL THIEM
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Triangular Matrix ◽  
Lower Triangular Matrix ◽  
Transition Matrices ◽  
Triangular Matrices ◽  
Power Sum ◽  
Upper Triangular Matrices ◽  
Upper Triangular Matrix ◽  
Lower Triangular

We give an LU-decomposition of the supercharacter table of the group of n × n unipotent upper triangular matrices over 𝔽q, into a lower-triangular matrix with entries in ℤ[q] and an upper-triangular matrix with entries in ℤ[q-1]. To this end, we introduce a q deformation of a new power-sum basis of the Hopf algebra of symmetric functions in noncommuting variables. The decomposition is obtained from the transition matrices between the supercharacter basis, the q-power-sum basis and the superclass basis. This is similar to the decomposition of the character table of the symmetric group Sn given by the transition matrices between Schur functions, monomials and power-sums. We deduce some combinatorial results associated to this decomposition. In particular, we compute the determinant of the supercharacter table.

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