Unipotent Upper-triangular Matrices Research Articles

Generalized Steinberg relations

We consider a field F and positive integers n, m, such that m is not divisible by $$\mathrm {Char}(F)$$ and is prime to n!. The absolute Galois group $$G_F$$ acts on the group $$\mathbb {U}_n(\mathbb {Z}/m)$$ of all $$(n+1)\times (n+1)$$ unipotent upper-triangular matrices over $$\mathbb {Z}/m$$ cyclotomically. Given $$0,1\ne z\in F$$ and an arbitrary list w of n Kummer elements $$(z)_F$$ , $$(1-z)_F$$ in $$H^1(G_F,\mu _m)$$ , we construct in a canonical way a quotient $$\mathbb {U}_w$$ of $$\mathbb {U}_n(\mathbb {Z}/m)$$ and a cohomology element $$\rho ^z$$ in $$H^1(G_F,\mathbb {U}_w)$$ whose projection to the superdiagonal is the prescribed list. This extends results by Wickelgren, and in the case $$n=2$$ recovers the Steinberg relation in Galois cohomology, proved by Tate.

Shell Tableaux: A Set Partition Analog of Vacillating Tableaux

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.

Normal supercharacter theories and their supercharacters

In order to find a tractable theory to substitute for the wild character theory of the group of n×n unipotent upper-triangular matrices over a finite field Fq, André and Yan introduced the notion of supercharacter theory. In this paper, we construct a supercharacter theory from an arbitrary set S of normal subgroups of G. We call such supercharacter theory the normal supercharacter theory generated by S. It is shown that normal supercharacter theories are integral, and a recursive formula for supercharacters of the normal supercharacter theory is provided. Also, we indicate that the superclasses of the normal supercharacter theory generated by all normal subgroups of G are given by certain values on the primitive central idempotents. We study the connection between the finest normal supercharacter theory and faithful irreducible characters. Moreover, an algorithm is presented to construct the supercharacter table of the finest normal supercharacter theory from the character table. Finally, we argue that normal supercharacter theories cannot be obtained by previously known supercharacter theory constructions.

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

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.

A SUPERCHARACTER TABLE DECOMPOSITION VIA POWER-SUM SYMMETRIC FUNCTIONS

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 Sngiven 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.

The number of conjugacy classes in pattern groups is not a polynomial function

Abstract A famous open problem due to Graham Higman asks if the number of conjugacy classes in the group of n × n unipotent upper triangular matrices over the q-element field can be expressed as a polynomial function of q for every fixed n. We consider the generalization of the problem for pattern groups and prove that for some pattern groups of nilpotency class two the number of conjugacy classes is not a polynomial function of q.

Elementary abelian Hopf Galois structures and polynomial formal groups

We find a relationship between regular embeddings of G, an elementary abelian p-group of order p n , into unipotent upper triangular matrices with entries in F p and commutative dimension n degree 2 polynomial formal groups with nilpotent upper triangular structure matrices. We classify the latter when n = 3 up to linear isomorphism, and use that classification to determine the number of Hopf Galois structures on a Galois extension L / K of fields with Galois group G elementary abelian of order p 3 . We also obtain a lower bound on the number of Hopf Galois structures on a Galois extension L / K when the Galois group is elementary abelian of order p n .