Monomial Groups Research Articles

Irreducible p-constant characters of finite reflection groups

Abstract A complex irreducible character of a finite group G is said to be p-constant, for some prime p dividing the order of G, if it takes constant value at the set of p-singular elements of G. In this paper we classify irreducible p-constant characters for finite reflection groups, nilpotent groups and complete monomial groups. We also propose some conjectures about the structure of the groups admitting such characters.

On the full monomial automorphism groups of Reed–Solomon codes and their MDS-extensions

We determine the full monomial automorphism groups of Reed–Solomon codes and of theirMDS-extensions by adding one, two, and three symbols.We show that several Reed–Solomon codes, extended by two symbols, are equivalent to constacyclic codes and, under certain conditions, to cyclic codes.

On the Representations of M-Groups

The main object of this paper is to study the representations of monomial groups and characters technique for representations of monomial groups. We refer to monomial groups by M-groups. Moreover we investigate the relation of monomial groups and solvable groups. Many applications have been given the symbol G e.g. group of order 297 is an M-group and solvable. For any group G, the factor group G/G? (G? is the derived subgroup of G) is an M-group in particular if G = Sn, SL(4,R).

Coleman Automorphisms of Permutational Wreath Products

Let N be a finite nontrivial nilpotent group and H a finite centerless permutation group on a finite set Ω (i.e., H acts faithfully on Ω). Let G = N≀H = N|Ω| ⋊ H be the corresponding permutational wreath product of N by H. It is shown that every Coleman automorphism of G is an inner automorphism. This generalizes a well-known result due to Petit Lobão and Sehgal stating that the normalizer property holds for complete monomial groups with nilpotent base groups.

Nice Reflection Arrangements

The aim of this note is a classification of all nice and all inductively factored reflection arrangements. It turns out that apart from the supersolvable instances only the monomial groups $G(r,r,3)$ for $r \ge 3$ give rise to nice reflection arrangements. As a consequence of this and of the classification of all inductively free reflection arrangements from Hoge and Röhrle (2015) we deduce that the class of all inductively factored reflection arrangements coincides with the class of all supersolvable reflection arrangements. Moreover, we extend these classifications to hereditarily factored and hereditarily inductively factored reflection arrangements.

On the Representations of M-Groups

The main object of this paper is to study the representations of monomial groups and characters technique for representations of monomial groups. We refer to monomial groups by M-groups. Moreover we investigate the relation of monomial groups and solvable groups. Many applications have been given the symbol G e.g. group of order 297 is an M-group and solvable. For any group G, the factor group G/G? (G? is the derived subgroup of G) is an M-group in particular if G = Sn, SL(4,R).

On Idempotents and the Number of Simple Components of Semisimple Group Algebras

We describe the primitive central idempotents of the group algebra over a number field of finite monomial groups. We give also a description of the Wedderburn decomposition of the group algebra over a number field for finite strongly monomial groups. Further, for this class of group algebras, we describe when the number of simple components agrees with the number of simple components of the rational group algebra. Finally, we give a formula for the rank of the central units of the group ring over the ring of integers of a number field for a strongly monomial group.

On the index of a free abelian subgroup in the group of central units of an integral group ring

Let Z(U(Z[G])) denote the group of central units in the integral group ring Z[G] of a finite group G. A bound on the index of the subgroup generated by a virtual basis in Z(U(Z[G])) is computed for a class of strongly monomial groups. The result is illustrated with application to the groups of order pn, p prime, n≤4. The rank of Z(U(Z[G])) and the Wedderburn decomposition of the rational group algebra of these p-groups have also been obtained.

Factoring Matrices into the Product of Circulant and Diagonal Matrices

A generic matrix $$A\in \,\mathbb {C}^{n \times n}$$ is shown to be the product of circulant and diagonal matrices with the number of factors being $$2n-1$$ at most. The demonstration is constructive, relying on first factoring matrix subspaces equivalent to polynomials in a permutation matrix over diagonal matrices into linear factors. For the linear factors, the sum of two scaled permutations is factored into the product of a circulant matrix and two diagonal matrices. Extending the monomial group, both low degree and sparse polynomials in a permutation matrix over diagonal matrices, together with their permutation equivalences, constitute a fundamental sparse matrix structure. Matrix analysis gets largely done polynomially, in terms of permutations only.

On non-abelian Brumer and Brumer–Stark conjecture for monomial CM-extensions

Let K/k be a finite Galois CM-extension of number fields whose Galois group G is monomial and S a finite set of places of k. Then the "Stickelberger element" θK/k,S is defined. Concerning this element, Andreas Nickel formulated the non-abelian Brumer and Brumer–Stark conjectures and their "weak" versions. In this paper, when G is a monomial group, we prove that the weak non-abelian conjectures are reduced to the weak conjectures for abelian subextensions. We write D4p, Q2n+2 and A4 for the dihedral group of order 4p for any odd prime p, the generalized quaternion group of order 2n+2 for any natural number n and the alternating group on four letters respectively. Suppose that G is isomorphic to D4p, Q2n+2 or A4 × ℤ/2ℤ. Then we prove the l-parts of the weak non-abelian conjectures, where l = 2 in the quaternion case, and l is an arbitrary prime which does not split in ℚ(ζp) in the dihedral case and in ℚ(ζ3) in the alternating case. In particular, we do not exclude the 2-part of the conjectures and do not assume that S contains all finite places which ramify in K/k in contrast with Nickel's formulation.

Central units of integral group rings

We give an explicit description for a basis of a subgroup of finite index in the group of central units of the integral group ring $\Z G$ of a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in $G$. The basis elements turn out to be a natural product of conjugates of Bass units. This extends and generalizes a result of Jespers, Parmenter and Sehgal showing that the Bass units generate a subgroup of finite index in the center $\mathcal{Z} (\U (\Z G))$ of the unit group $\U (\Z G)$ in case $G$ is a finite nilpotent group. Next, we give a new construction of units that generate a subgroup of finite index in $\mathcal{Z}(\U(\Z G))$ for all finite strongly monomial groups $G$. We call these units generalized Bass units. Finally, we show that the commutator group $\U(\Z G)/\U(\Z G)'$ and $\mathcal{Z}(\U(\Z G))$ have the same rank if $G$ is a finite group such that $\Q G$ has no epimorphic image which is either a non-commutative division algebra other than a totally definite quaternion algebra, or a two-by-two matrix algebra over a division algebra with center either the rationals or a quadratic imaginary extension of $\Q$. This allows us to prove that in this case the natural images of the Bass units of $\Z G$ generate a subgroup of finite index in $\U(\Z G)/\U(\Z G)'$.

The Normalizer Property for Integral Group Rings of a Class of Complete Monomial Groups

Let G = H≀S m be the natural wreath product of H by S m , where H is a finite 2-closed group and S m is the symmetric group of degree m. It is shown that the normalizer property holds for G.

The rational group algebra of a normally monomial group

In this paper, a complete irredundant set of a class of strong Shoda pairs of a finite group G is computed. The algebraic structure of the rational group algebra of a normally monomial group is thus obtained. A necessary and sufficient condition for G to be normally monomial is derived. The main result is also illustrated by computing a complete set of primitive central idempotents and the explicit Wedderburn decomposition of the rational group algebra of some normally monomial groups.

Group rings of finite strongly monomial groups: Central units and primitive idempotents

We compute the rank of the group of central units in the integral group ring ZG of a finite strongly monomial group G. The formula obtained is in terms of the strong Shoda pairs of G. Next we construct a virtual basis of the group of central units of ZG for a class of groups G properly contained in the finite strongly monomial groups. Furthermore, for another class of groups G inside the finite strongly monomial groups, we give an explicit construction of a complete set of orthogonal primitive idempotents of QG.Finally, we apply these results to describe finitely many generators of a subgroup of finite index in the group of units of ZG, this for metacyclic groups G of the form G=Cqm⋊Cpn with p and q different primes and the cyclic group Cpn of order pn acting faithfully on the cyclic group Cqm of order qm.

Automorphism groups of some pure braid groups

We find finite presentations for the automorphism group of the Artin pure braid group and the automorphism group of the pure braid group associated to the full monomial group.

The rationality problem for purely monomial group actions

We investigate the rationality problem for purely monomial actions of finite groups. We solve it affirmatively in the following case: K is a field with char K ≠ 2 and G is a subgroup of GL(n; ℤ) isomorphic to (C 2 ), where n > 0. Then the fixed field of K(x 1, ... x n ) under the purely monomial action of G is rational over K.

Combinatorics of Cremona monomial maps

One studies Cremona monomial maps by combinatorial means. Among the results is a simple integer matrix theoretic proof that the inverse of a Cremona monomial map is also defined by monomials of fixed degree, and moreover, the set of monomials defining the inverse can be obtained explicitly in terms of the initial data. A neat consequence is drawn for the plane Cremona monomial group, in particular the known result saying that a plane Cremona (monomial) map and its inverse have the same degree. Included is a discussion about the computational side and/or implementation of the combinatorial invariants stemming from these questions.

On Representation of Monomial Groups

Taketa shows that all monomial groups (commonly written as M -groups) are solvable. Gajendragadkar gives the notion of π-factorable character. We show that an irreducible character of an M-group is primitive if it is π-factorable. Issacs proves that product of two monomial characters is a monomial. We extend this fact to include any finite number of monomial characters consequently we prove that any product of finite number of M-groups is an M-group. We show that any group of order 45 is an M-group and for any group G, the factor group /

Rationality problem of three-dimensional monomial group actions

Let K be a field of characteristic not two and K ( x , y , z ) the rational function field over K with three variables x , y , z . Let G be a finite group acting on K ( x , y , z ) by monomial K-automorphisms. We consider the rationality problem of the fixed field K ( x , y , z ) G under the action of G, namely whether K ( x , y , z ) G is rational (that is, purely transcendental) over K or not. We may assume that G is a subgroup of GL ( 3 , Z ) and the problem is determined up to conjugacy in GL ( 3 , Z ) . There are 73 conjugacy classes of G in GL ( 3 , Z ) . By results of Endo–Miyata, Voskresenskiĭ, Lenstra, Saltman, Hajja, Kang and Yamasaki, 8 conjugacy classes of 2-groups in GL ( 3 , Z ) have negative answers to the problem under certain monomial actions over some base field K, and the necessary and sufficient condition for the rationality of K ( x , y , z ) G over K is given. In this paper, we show that the fixed field K ( x , y , z ) G under monomial action of G is rational over K except for possibly negative 8 cases of 2-groups and unknown one case of the alternating group of degree four. Moreover we give explicit transcendental bases of the fixed fields over K. For the unknown case, we obtain an affirmative solution to the problem under some conditions. In particular, we show that if K is quadratically closed field then K ( x , y , z ) G is rational over K. We also give an application of the result to 4-dimensional linear Noetherʼs problem.

Generic Hecke algebras for monomial groups

In this paper we define a two-variable, generic Hecke algebra,H\mathcal H, for each complex reflection groupG(b,1,n)G(b,1,n). The algebraH\mathcal Hspecializes to the group algebra ofG(b,1,n)G(b,1,n)and also to an endomorphism algebra of a representation ofGLn⁡(Fq)\operatorname {GL}_n(\mathbb F_q)induced from a solvable subgroup. We construct Kazhdan-Lusztig “RR-polynomials” forH\mathcal {H}and show that they may be used to define a partial order onG(b,1,n)G(b,1,n). Using a generalization of Deodhar’s notion of distinguished subexpressions we give a closed formula for theRR-polynomials. After passing to a one-variable quotient of the ring of scalars, we construct Kazhdan-Lusztig polynomials forH\mathcal Hthat reduce to the usual Kazhdan-Lusztig polynomials for the symmetric group whenb=1b=1.