Irreducible Group Research Articles

Regular theory in complex braid groups

In his seminal paper [2], Bessis introduces a Garside structure for the braid group of a well-generated irreducible complex reflection group. Using this Garside structure, he establishes a strong connection between regular elements in the reflection group, and roots of the “full twist” element of the pure braid group.He then suggests that it would be possible to extend the conclusion of this theorem to centralizers of regular elements in well-generated groups. In this paper we give a positive answer to this question and we show moreover that these results hold for an arbitrary reflection group. As a byproduct, we get a generalization of a theorem from Shvartsman regarding the torsion of the quotient of an irreducible braid group by its center.

The significance of GDP: a new take on a century-old question

ABSTRACT What is the significance of GDP per capita to a society? What does it represent conceptually? These questions have been addressed in past decades, engendering extensive explorations of the limitations of the indicator, yet answers have proved problematic or partial. The paper presents the main conclusions so far drawn and builds upon them to present a new reading of the significance of GDP per capita. At the heart of this reading is the view that, while GDP per capita is not indicative of the welfare of individuals, it is indicative of an irreducible ‘group well-being.’ This view, however, requires one to relinquish the belief that only individuals can be ‘well.’ It turns out that the allegedly orthodox view, which sees GDP as a human-centered indicator, requires an unorthodox philosophical standpoint, one that accepts an irreducible group well-being. The paper presents this alternative interpretation and addresses its upsides and limitations.

О базисных инвариантах некоторых конечных подгрупп в SL3(C)

The work is devoted to the study of algebras of invariants of finite unitary groups G' = G∩SL3(C), where G is a finite unitary irreducible primitive group generated by reflections in the unitary space U3. It is known that the system of invariants of the group G' that form an algebra is obtained from the system of invariants of the group G that form the algebra by adding all semi-invariants of the group G of a special form. In the paper, generators of the algebras of invariants of all the indicated groups G' are constructed.

Lattice associated to a Shi variety

Let $W$ be a irreducible Weyl group and $W_a$ its affine Weyl group. In a previous article the author defined an affine variety $\widehat{X}_{W_a}$, called the Shi variety of $W_a$, whose integral points are in bijection with $W_a$. The set of irreducible components of $\widehat{X}_{W_a}$, denoted $H^0(\widehat{X}_{W_a})$, is of some interest and we show in this article that $H^0(\widehat{X}_{W_a})$ has a structure of semidistributive lattice.

Steinberg's fixed point theorem for crystallographic complex reflection groups

Steinberg's fixed point theorem states that given a finite complex reflection group the stabilizer subgroup of a point is generated by reflections that fix this point. This statement is also true for affine Weyl groups. Of the infinite discrete complex reflection groups, it was shown that there are some infinite complex reflections groups that have non-trivial stabilizers that do not contain a single reflection, and therefore, these groups cannot satisfy the fixed point theorem. We thus classify the infinite discrete irreducible complex reflection groups of the infinite family which satisfy the statement of the fixed point theorem.

Perfect models for finite Coxeter groups

A model for a finite group is a set of linear characters of subgroups that can be induced to obtain every irreducible character exactly once. A perfect model for a finite Coxeter group is a model in which the relevant subgroups are the quasiparabolic centralizers of perfect involutions. In prior work, we showed that perfect models give rise to interesting examples of W-graphs. Here, we classify which finite Coxeter groups have perfect models. Specifically, we prove that the irreducible finite Coxeter groups with perfect models are those of types An, Bn, D2n+1, H3, or I2(n). We also show that up to a natural form of equivalence, outside types A3, Bn, and H3, each irreducible finite Coxeter group has at most one perfect model. Along the way, we also prove a technical result about representations of finite Coxeter groups, namely, that induction from standard parabolic subgroups of corank at least two is never multiplicity-free.

Coxeter pairs, Ammann patterns, and Penrose-like tilings

We identify a precise geometric relationship between: (i) certain natural pairs of irreducible reflection groups (``Coxeter pairs"); (ii) self-similar quasicrystalline patterns formed by superposing sets of 1D quasi-periodically-spaced lines, planes or hyper-planes (``Ammann patterns"); and (iii) the tilings dual to these patterns (``Penrose-like tilings"). We use this relationship to obtain all irreducible Ammann patterns and their dual Penrose-like tilings, along with their key properties in a simple, systematic and unified way, expanding the number of known examples from four to infinity. For each symmetry, we identify the minimal Ammann patterns (those composed of the fewest 1d quasiperiodic sets) and construct the associated Penrose-like tilings: 11 in 2D, 9 in 3D and one in 4D. These include the original Penrose tiling, the four other previously known Penrose-like tilings, and sixteen that are new. We also complete the enumeration of the quasicrystallographic space groups corresponding to the irreducible non-crystallographic reflection groups, by showing that there is a unique such space group in 4D (with nothing beyond 4D).

Pop-stack-sorting for Coxeter groups

Let \(W\) be an irreducible Coxeter group. We define the Coxeter pop-stack-sorting operator \(\mathsf{Pop}:W\to W\) to be the map that fixes the identity element and sends each nonidentity element \(w\) to the meet of the elements covered by \(w\) in the right weak order. When \(W\) is the symmetric group \(S_n\), \(\mathsf{Pop}\) coincides with the pop-stack-sorting map. Generalizing a theorem about the pop-stack-sorting map due to Ungar, we prove that \[\sup\limits_{w\in W}\left|O_{\mathsf{Pop}}(w)\right|=h,\] where \(h\) is the Coxeter number of \(W\) (with \(h=\infty\) if \(W\) is infinite) and \(O_f(w)\) denotes the forward orbit of \(w\) under a map \(f\). When \(W\) is finite, this result is equivalent to the statement that the maximum number of terms appearing in the Brieskorn normal form of an element of \(W\) is \(h-1\). More generally, we define a map \(f:W\to W\) to be compulsive if for every \(w\in W\), \(f(w)\) is less than or equal to \(\mathsf{Pop}(w)\) in the right weak order. We prove that if \(f\) is compulsive, then \(\sup\limits_{w\in W}|O_f(w)|\leq h\). This result is new even for symmetric groups. We prove that \(2\)-pop-stack-sortable elements in type \(B\) are in bijection with \(2\)-pop-stack-sortable permutations in type \(A\), which were enumerated by Pudwell and Smith. Claesson and Gu{\dh}mundsson proved that for each fixed nonnegative integer \(t\), the generating function that counts \(t\)-pop-stack-sortable permutations in type \(A\) is rational; we establish analogous results in types \(B\) and \(\widetilde A\).Mathematics Subject Classifications: 05E16, 37E15, 05A05Keywords: Pop-stack-sorting, Coxeter group, weak order, Coxeter number, compulsive map, regular language

Frobenius Manifolds on Orbits Spaces

The orbits space of an irreducible linear representation of a finite group is a variety whose coordinate ring is the ring of invariant polynomials. Boris Dubrovin proved that the orbits space of the standard reflection representation of an irreducible finite Coxeter group \({\mathcal {W}}\) acquires a natural polynomial Frobenius manifold structure. We apply Dubrovin’s method on various orbits spaces of linear representations of finite groups. We find some of them has non or several natural Frobenius manifold structures. On the other hand, these Frobenius manifold structures include rational and trivial structures which are not known to be related to the invariant theory of finite groups.

Recognition of absolutely irreducible matrix groups that are tensor decomposable or induced

Let V be an irreducible module for a finite group G over an algebraically closed field k‾. We prove that the algebra Homk‾(V,V) has a proper G-invariant subalgebra if and only if V is either imprimitive or tensor decomposable. We give an algorithm to determine whether an absolutely irreducible matrix representation defined over a finite field has one of these properties. Our algorithm reduces the computation to an instance of the Pure Tensor problem. This asks whether a subspace of a tensor product of vector spaces X and Y contains an element of the form x⊗y with x∈X and y∈Y. We show that the Pure Tensor Problem reduces to the calculation of an appropriate Gröbner basis.

Maximal order Abelian subgroups of Coxeter groups

AbstractIn this note, we give a classification of the maximal order Abelian subgroups of finite irreducible Coxeter groups. We also prove a Weyl group analog of Cartan’s theorem that all maximal tori in a connected compact Lie group are conjugate.

Coxeter Pop-Tsack Torsing

Given a finite irreducible Coxeter group W with a fixed Coxeter element c, we define the Coxeter pop-tsack torsing operator Pop T :W→W by Pop T (w)=w·π T (w) -1 , where π T (w) is the join in the noncrossing partition lattice NC(w,c) of the set of reflections lying weakly below w in the absolute order. This definition serves as a “Bessis dual” version of the first author’s notion of a Coxeter pop-stack sorting operator, which, in turn, generalizes the pop-stack sorting map on symmetric groups. We show that if W is coincidental or of type D, then the identity element of W is the unique periodic point of Pop T and the maximum size of a forward orbit of Pop T is the Coxeter number h of W. In each of these types, we obtain a natural lift from W to the dual braid monoid of W. We also prove that W is coincidental if and only if it has a unique forward orbit of size h. For arbitrary W, we show that the forward orbit of c -1 under Pop T has size h and is isolated in the sense that none of the non-identity elements of the orbit have preimages lying outside of the orbit.

Ricci curvature of Bruhat orders

We study the Ricci curvature of the Hasse diagrams of the Bruhat order of finite irreducible Coxeter groups. For this purpose we compute the maximum degree of these graphs for types Bn and Dn. The proof uses a new graph Γ(π) defined for any element π in the corresponding group.

Thin right‐angled Coxeter groups in some uniform arithmetic lattices

Using a variant of an unpublished argument due to Agol, we show that an irreducible right-angled Coxeter group on n ⩾ 3 ${n \geqslant 3}$ vertices embeds as a thin subgroup of a uniform arithmetic lattice in an indefinite orthogonal group O ( p , q ) $\mathrm{O}(p, q)$ for some p , q ⩾ 1 $p, q \geqslant 1$ satisfying p + q = n $p+q=n$ .

Comparing the order and the minimal number of generators of a finite irreducible linear group

We denote the smallest size of a generating set of a group G by d(G). We prove that d(G)log⁡|G|=O(n2log⁡q) for irreducible subgroups G of GL(n,q), and estimate the associated constants. The result is motivated by attempts to bound the complexity of computing the automorphism groups of various classes of finite groups.

The centralizer of a Coxeter element

We prove that the centralizer of a Coxeter element in an irreducible Coxeter group is the cyclic group generated by that Coxeter element.

Lauricella's $F_C$ with finite irreducible monodromy group

This paper presents our study of the conditions under which the monodromy group for Lauricella's hypergeometric function $F_C (a,b,c;x)$ is finite irreducible. We provide these conditions in terms of the parameters $a, b, c$. In addition, we discuss the structure of the finite irreducible monodromy group.

Principal series of Hermitian Lie groups induced from Heisenberg parabolic subgroups

Let G be an irreducible Hermitian Lie group and D=G/K its bounded symmetric domain in Cd of rank r. Each γ of the Harish-Chandra strongly orthogonal roots {γ1,⋯,γr} defines a Heisenberg parabolic subgroup P=MAN of G. We study the principal series representations IndPG(1⊗eν⊗1) of G induced from P. These representations can be realized as the L2-space on the minimal K-orbit S=Ke=K/L of a root vector e of γ in Cd, and S is a circle bundle over a compact Hermitian symmetric space K/L0 of K of rank one or two. We find the complementary series, reduction points, and unitary sub-quotients in this family of representations.

Acylindrical hyperbolicity for Artin groups of dimension 2

In this paper, we show that every irreducible 2-dimensional Artin group A_{Gamma } of rank at least 3 is acylindrically hyperbolic. We do this by studying the action of A_Gamma on its modified Deligne complex. Along the way, we prove results of independent interests on the geometry of links of this complex.

Minimizing capacity among linear images of rotationally invariant conductors

Logarithmic capacity is shown to be minimal for a planar set having N-fold rotational symmetry (\(N \ge 3\)), among all conductors obtained from the set by area-preserving linear transformations. Newtonian and Riesz capacities obey a similar property in all dimensions, when suitably normalized linear transformations are applied to a set having irreducible symmetry group. A corollary is Pólya and Schiffer’s lower bound on capacity in terms of moment of inertia.