Unitriangular Group Research Articles

INVARIANTS OF THE COADJOINT ACTION ON THE BASIC VARIETIES OF THE UNITRIANGULAR GROUP

We find the generators of the fields of invariants of the coadjoint action of the unitriangular group on the basic varieties and basic cells. It is proved that the transcendental degree of the field of invariants on a basic cell coincides with the number of factors in the special factorization of the associated element of the Weyl group as a product of reflections.

Schur multipliers of unitriangular groups

An explicit formula for the Schur multiplier of the group of unitriangular matrices over products of cyclic rings Z/mZ and Z is derived. We use it to provide presentations of the corresponding covering groups and touch upon the question of finding their nonisomorphic representatives.

The local structure of groups of triangular automorphisms of relatively free algebras

Let K be an arbitrary field and C n a relatively free algebra of rank n. In particular, as C n we may treat a polynomial algebra P n , a free associative algebra A n , or an absolutely free algebra F n . For the algebras C n = P n , A n , F n , it is proved that every finitely generated subgroup G of a group TC n of triangular automorphisms admits a faithful matrix representation over a field K; hence it is residually finite by Mal’tsev’s theorem. For any algebra C n , the triangular automorphism group TC n is locally soluble, while the unitriangular automorphism group UC n is locally nilpotent. Consequently, UC n is local (linear and residually finite). Also it is stated that the width of the commutator subgroup of a finitely generated subgroup G of UC n can be arbitrarily large with increasing n or transcendence degree of a field K over the prime subfield.

CLASS PRESERVING AUTOMORPHISMS OF UNITRIANGULAR GROUPS

Let UTn(K) be a unitriangular group over a field K. We prove that the group of all class preserving automorphisms of UTn(K) is equal to Inn ( UTn(K)) if and only if K is a prime field. Let [Formula: see text], where γ3( UTn(𝔽pm)) denotes the third term of the lower central series of UTn(𝔽pm). We calculate the group of all class preserving automorphisms and class preserving outer automorphisms of [Formula: see text].

Actions and Identities on Set Partitions

A labeled set partition is a partition of a set of integers whose arcs are labeled by nonzero elements of an abelian group $\mathbb{A}$. Inspired by the action of the linear characters of the unitriangular group on its supercharacters, we define a group action of $\mathbb{A}^n$ on the set of $\mathbb{A}$-labeled partitions of an $(n+1)$-set. By investigating the orbit decomposition of various families of set partitions under this action, we derive new combinatorial proofs of Coker's identity for the Narayana polynomial and its type B analogue, and establish a number of other related identities. In return, we also prove some enumerative results concerning André and Neto's supercharacter theories of type B and D.

Combinatorial methods of character enumeration for the unitriangular group

Let UT n ( q ) denote the group of unipotent n × n upper triangular matrices over a field with q elements. The degrees of the complex irreducible characters of UT n ( q ) are precisely the integers q e with 0 ⩽ e ⩽ ⌊ n 2 ⌋ ⌊ n − 1 2 ⌋ , and it has been conjectured that the number of irreducible characters of UT n ( q ) with degree q e is a polynomial in q − 1 with nonnegative integer coefficients (depending on n and e). We confirm this conjecture when e ⩽ 8 and n is arbitrary by a computer calculation. In particular, we describe an algorithm which allows us to derive explicit bivariate polynomials in n and q giving the number of irreducible characters of UT n ( q ) with degree q e when n > 2 e and e ⩽ 8 . When divided by q n − e − 2 and written in terms of the variables n − 2 e − 1 and q − 1 , these functions are actually bivariate polynomials with nonnegative integer coefficients, suggesting an even stronger conjecture concerning such character counts. As an application of these calculations, we are able to show that all irreducible characters of UT n ( q ) with degree ⩽ q 8 are Kirillov functions. We also discuss some related results concerning the problem of counting the irreducible constituents of individual supercharacters of UT n ( q ) .

Iterative character constructions for algebra groups

We construct a family of orthogonal characters of an algebra group which decompose the supercharacters defined by Diaconis and Isaacs (2008) [6]. Like supercharacters, these characters are given by nonnegative integer linear combinations of Kirillov functions and are induced from linear supercharacters of certain algebra subgroups. We derive a formula for these characters and give a condition for their irreducibility; generalizing a theorem of Otto (2010) [20], we also show that each such character has the same number of Kirillov functions and irreducible characters as constituents. In proving these results, we observe as an application how a recent computation by Evseev (2010) [7] implies that every irreducible character of the unitriangular group UT n ( q ) of unipotent n × n upper triangular matrices over a finite field with q elements is a Kirillov function if and only if n ⩽ 12 . As a further application, we discuss some more general conditions showing that Kirillov functions are characters, and describe some results related to counting the irreducible constituents of supercharacters.

Exotic characters of unitriangular matrix groups

Let UT n ( q ) denote the unitriangular group of unipotent n × n upper triangular matrices over a finite field with cardinality q and prime characteristic p . It has been known for some time that when p is fixed and n is sufficiently large, UT n ( q ) has “exotic” irreducible characters taking values outside the cyclotomic field Q ( ζ p ) . However, all proofs of this fact to date have been both non-constructive and computer dependent. In the preliminary work Marberg (2010) [15], we defined a family of orthogonal characters decomposing the supercharacters of an arbitrary algebra group. By applying this construction to the unitriangular group, we are able to derive by hand an explicit description of a family of characters of UT n ( q ) taking values in arbitrarily large cyclotomic fields. In particular, we prove that if r is a positive integer power of p and n > 6 r , then UT n ( q ) has an irreducible character of degree q 5 r 2 − 2 r which takes values outside Q ( ζ p r ) . By the same techniques, we are also able to construct explicit Kirillov functions which fail to be characters of UT n ( q ) when n > 12 and q is arbitrary.

The invariant field of the adjoint action of the unitriangular group in the nilradical of a parabolic subalgebra

In the present paper the invariant field of the adjoint action of the unitriangular group in the nilradical of any parabolic subalgebra is described. Bibliography: 7 titles.

Counting characters of small degree in upper unitriangular groups

Let U n denote the group of upper n × n unitriangular matrices over a fixed finite field F of order q . That is, U n consists of upper triangular n × n matrices having every diagonal entry equal to 1 . It is known that the degrees of all irreducible complex characters of U n are powers of q . It was conjectured by Lehrer that the number of irreducible characters of U n of degree q e is an integer polynomial in q depending only on e and n . We show that there exist recursive (for n ) formulas that this number satisfies when e is one of 1 , 2 and 3 , and thus show that the conjecture is true in those cases.

The group of permutation automorphisms of a q-ary hamming code

We prove that the group of permutation automorphism of a q-ary Hamming code of length n = (q m ? 1)/(q ? 1) is isomorphic to the unitriangular group UT m (q) if the code has a parity-check matrix composed of all columns of the form (0 ...0 1 * ... *)T. We also show that the group of permutation automorphisms of a cyclic Hamming code cannot be isomorphic to UT m (q). We thus show that equivalent codes can have different permutation automorphism groups.

Supercharacters of the Sylowp-subgroups of the finite symplectic and orthogonal groups

We define and study supercharacters of the classical finite unipotent groups of types Bn(q), Cn(q) and Dn(q). We show that the results we proved in 2006 remain valid over any finite field of odd characteristic. In particular, we show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters. In addition, we prove that the unitary vector space spanned by all the supercharacters is closed under multiplication, and establish a formula for the supercharacter values. As a consequence, we obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we give a combinatorial description of all the irreducible characters of maximum degree in terms of the root system, by showing how they can be obtained as constituents of particular supercharacters.

Subregular characters of the unitriangular group over a finite field

We prove a formula for subregular characters of the unitriangular group over a finite field in terms of coefficients of minors of the characteristic matrix.

Coadjoint orbits of the group UT(7, K)

We classify the irreducible representations and the coadjoint orbits of a unitriangular group of size less than or equal to seven. We classify the subregular orbits of a unitriangular group of arbitrary size.

Subgroups of unitriangular groups of infinite matrices

We show that for any associative ring R, the subgroup UTr(∞, R) of row-finite matrices in UT(∞, R), the group of all infinite-dimensional (indexed by ℕ) upper unitriangular matrices over R, is generated by the so-called strings (block-diagonal matrices with finite blocks along the main diagonal). This allows us to define a large family of subgroups of UTr(∞, R) associated with some growth functions. The smallest subgroup in this family, called the group of banded matrices, is generated by 1-banded simultaneous elementary transvections (a slight generalization of the usual notion of elementary transvection). We introduce the notion of net subgroup and characterize the normal net subgroups of UT(∞, R). Bibliography: 26 titles.

Super-characters of finite unipotent groups of types [formula omitted], [formula omitted] and [formula omitted

We define and study super-characters (over the complex field) of the classical finite unipotent groups of types B n , C n and D n . Under the assumption that the prime is sufficiently large, we extend the known results for the unitriangular group proved by the first author in the papers: [C.A.M. André, Basic characters of the unitriangular group, J. Algebra 175 (1995) 287–319], and [C.A.M. André, Basic characters of the unitriangular group (for arbitrary primes), Proc. Amer. Math. Soc. 130 (7) (2002) 1943–1954]. In particular, we prove that every irreducible (complex) character occurs as a constituent of a unique super-character. We also give a combinatorial description of all the irreducible characters of maximum degree.

TREE-WREATHING APPLIED TO GENERATION OF GROUPS BY FINITE AUTOMATA

We extend the tree-wreath product of groups introduced by Brunner and the author, as a generalization of the restricted wreath product, thus enlarging the class of groups known to be generated by finite synchronous automata. In particular, we prove that given a countable abelian residually finite 2 -group H and B = B(n,ℤ), a canonical subgroup of finite index in GL(n,ℤ), then the restricted wreath product H wr B can be generated by finite synchronous automata on 0,1. This is obtained by producing a representation of B as a group of automorphisms of the binary tree such that the stabilizer of the infinite sequence of 0's is trivial. The uni-triangular group U = U(n,ℤ) is a subgroup of B(n,ℤ) and so, H wr U also can be generated by finite synchronous automata on 0,1.

Isomorphisms of the Unitriangular Groups and Associated Lie Rings for the Exceptional Dimensions

Isomorphisms between finitary unitriangular groups and those of associated Lie rings are studied. In this paper we investigate exceptional cases.

Sylow Subgroups of the Chevalley Groups and Associated (Weakly) Finitary Groups and Rings

Automorphisms, isomorphisms and structural descriptions of finitary unitriangular groups over a ring and of Sylow subgroups of Chevalley groups are studied. Also, the theorem on pair unipotent intersections of Chevalley groups has been proved; it is connected with the known question on pair Sylow intersections of finite groups. These investigations use close connections and structural descriptions of considered groups and associated rings, which have been found recently in the case of a commutative ring of coefficients with a strongly maximal ideal. We consider some properties and examples of such ideals.

Constructive Matrix and Orderable Groups

We study into the relationship between constructivizations of an associative commutative ring K with unity and constructivizations of matrix groups GLn(K) (general), SLn(K) (special), and UTn(K) (unitriangular) over K. It is proved that for n≥ 3, a corresponding group is constructible iff so is K. We also look at constructivizations of ordered groups. It turns out that a torsion-free constructible Abelian group is orderly constructible. It is stated that the unitriangular matrix group UTn(K) over an orderly constructible commutative associative ring K is itself orderly constructible. A similar statement holds also for finitely generated nilpotent groups, and countable free nilpotent groups.