Real Reflection Group Research Articles

A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration of that module. Our approach is based on a detailed analysis of a flat connection applied to the primitive vector field. This generalizes and unifies analogous results for real reflection groups.

Counting nearest faraway flats for Coxeter chambers

In a finite Coxeter group W and with two given conjugacy classes of parabolic subgroups [X] and [Y] , we count those parabolic subgroups of W in [Y] that are full support, while simultaneously being simple extensions (i.e., extensions by a single reflection) of some standard parabolic subgroup of W in [X] . The enumeration is given by a product formula that depends only on the two parabolic types. Our derivation is case-free and combines a geometric interpretation of the “full support” property with a double-counting argument involving Crapo's beta invariant. As a corollary, this approach gives the first case-free proof of Chapoton's formula for the number of reflections of full support in a real reflection group W .

The freeness and trace conjectures for parabolic Hecke subalgebras

The two most fundamental conjectures on the structure of the generic Hecke algebra $\mathcal{H}(W)$ associated with a complex reflection group $W$ state that $\mathcal{H}(W)$ is a free module of rank $|W|$ over its ring of definition, and that $\mathcal{H}(W)$ admits a canonical symmetrising trace. The first conjecture has recently become a theorem, while the second conjecture, known to hold for real reflection groups, has only been proved for some exceptional non-real complex reflection groups (all of rank $2$ but one). The two most fundamental conjectures on the structure of the parabolic Hecke subalgebra $\mathcal{H}(W')$ associated with a parabolic subgroup $W'$ of $W$ state that $\mathcal{H}(W)$ is a free left and right $\mathcal{H}(W')$-module of rank $|W|/|W'|$, and that the canonical symmetrising trace of $\mathcal{H}(W')$ is the restriction of the canonical symmetrising trace of $\mathcal{H}(W)$ to $\mathcal{H}(W')$. Until now, these two conjectures have only be known to be true for real reflection groups. We prove them for all complex reflection groups of rank $2$ for which the BMM symmetrising trace conjecture is known to hold.

The Formal Affine Demazure Algebra and Real Finite Reflection Groups

In this paper, we generalize the formal affine Demazure algebra of Hoffnung-Malagón-López-Savage-Zainoulline to all real finite reflection groups. We begin by generalizing the formal group ring of Calmès-Petrov-Zainoulline to all real finite reflection groups. We then define and study the formal Demazure operators that act on the formal group ring. Using these results and constructions, we define and study the formal affine Demazure algebra for all real finite reflection groups. Finally, we compute several structure coefficients that appear in a braid relation among the formal Demazure elements, and we conclude this paper by computing all structure coefficients for the reflection groups I2(5), I2(7), H3, and H4.

Cellularity of endomorphism algebras of tilting objects

We show that, in a highest weight category with duality, the endomorphism algebra of a tilting object is naturally a cellular algebra. Our proof generalizes a recent construction of Andersen, Stroppel, and Tubbenhauer [4]. This result raises the question of whether all cellular algebras can be realized in this way. The construction also works without the presence of a duality and yields standard bases, in the sense of Du and Rui, which have similar combinatorial features to cellular bases. As an application, we obtain standard bases—and thus a general theory of “cell modules”—for Hecke algebras associated to finite complex reflection groups (as introduced by Broué, Malle, and Rouquier) via category O of the rational Cherednik algebra. For real reflection groups these bases are cellular.

Counting chains in the noncrossing partition lattice via the W-Laplacian

We give an elementary, case-free, Coxeter-theoretic derivation of the formula hnn!/|W| for the number of maximal chains in the noncrossing partition lattice NC(W) of a real reflection group W. Our proof proceeds by comparing the Deligne-Reading recursion with a parabolic recursion for the characteristic polynomial of the W-Laplacian matrix considered in our previous work. We further discuss the consequences of this formula for the geometric group theory of spherical and affine Artin groups.

Eulerian representations for real reflection groups

The Eulerian idempotents, first introduced for the symmetric group and later extended to all reflection groups, generate a family of representations called the Eulerian representations that decompose the regular representation. In Type $A$, the Eulerian representations have many elegant but mysterious connections to rings naturally associated with the braid arrangement. In this paper, we unify these results and show that they hold for any reflection group of coincidental type -- that is, $S_{n}$, $B_{n}$, $H_{3}$ or the dihedral group $I_{2}(m)$ -- by giving six characterizations of the Eulerian representations, including as components of the associated graded of the Varchenko-Gelfand ring $\mathcal{V}$. As a consequence, we show that Solomon's descent algebra contains a commutative subalgebra generated by sums of elements with the same number of descents if and only if $W$ is coincidental. More generally, when $W$ is any finite real reflection group, we give a case-free construction of a family of Eulerian representations described by a flat-decomposition of the ring $\mathcal{V}$.

Invariant theory for coincidental complex reflection groups

V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and he speculated that it had a certain product formula involving the exponents of the group. We show that Molchanov’s speculation is false in general but holds for all coincidental complex reflection groups when appropriately modified using exponents and co-exponents. These are the irreducible well-generated (i.e., duality) reflection groups with exponents forming an arithmetic progression and include many real reflection groups and all non-real Shephard groups, e.g., the Shephard-Todd infinite family G(d, 1, n). We highlight consequences for the q-Narayana and q-Kirkman polynomials, giving simple product formulas for both, and give a q-analogue of the identity transforming the h-vector to the f-vector for the coincidental finite type cluster/Cambrian complexes of Fomin–Zelevinsky and Reading. We include the determination of the Hilbert series for the non-coincidental irreducible complex reflection groups as well.

A factorisation theorem for the coinvariant algebra of a unitary reflection group

We prove the following theorem. Let G be a finite group generated by unitary reflections in a complex Hermitian space $$V={\mathbb {C}}^\ell $$ and let $$G'$$ be any reflection subgroup of G. Let $${\mathcal {H}}={\mathcal {H}}(G)$$ be the space of G-harmonic polynomials on V. There is a degree preserving isomorphism $$\mu :{\mathcal {H}}(G')\otimes {\mathcal {H}}(G)^{G'}\overset{\sim }{{\longrightarrow \;}}{\mathcal {H}}(G)$$ of graded $${\mathcal {N}}$$-modules, where $${\mathcal {N}}:=N_{{\text {GL}}(V)}(G)\cap N_{{\text {GL}}(V)}(G')$$ and $${\mathcal {H}}(G)^{G'}$$ is the space of $$G'$$-fixed points of $${\mathcal {H}}(G)$$. This generalises a result of Douglass and Dyer for parabolic subgroups of real reflection groups. An application is given to counting rational conjugates of reductive groups over $${\mathbb {F}}_q$$.

Reflection groups and 3d mathcal{N} > 6 SCFTs

We point out that the moduli spaces of all known 3d mathcal{N} = 8 and mathcal{N} = 6 SCFTs, after suitable gaugings of finite symmetry groups, have the form ℂ4r/Γ where Γ is a real or complex reflection group depending on whether the theory is mathcal{N} = 8 or mathcal{N} = 6, respectively. Real reflection groups are either dihedral groups, Weyl groups, or two sporadic cases H3,4 Since the BLG theories and the maximally supersymmetric Yang-Mills theories correspond to dihedral and Weyl groups, it is strongly suggested that there are two yet-to­be-discovered 3d mathcal{N} = 8 theories for H3,4. We also show that all known mathcal{N} = 6 theories correspond to complex reflection groups collectively known as G(k, x, N). Along the way, we demonstrate that two ABJM theories (SU(N)k x SU(N)-k)/ℤN and (U(N)k x U(N)-k) /ℤk are actually equivalent.

A rotation group whose subspace arrangement is not from a real reflection group

We exhibit a family of real rotation groups whose subspace arrangements are not contained in that of any real reflection group, answering a question of Martino and Singh.

The topology of arrangements of ideal type

In 1962, Fadell and Neuwirth showed that the configuration space of the braid arrangement is aspherical. Having generalized this to many real reflection groups, Brieskorn conjectured this for all finite Coxeter groups. This in turn follows from Deligne’s seminal work from 1972, where he showed that the complexification of every real simplicial arrangement is a K ( π , 1 ) –arrangement. We study the K ( π , 1 ) –property for a certain class of subarrangements of Weyl arrangements, the so-called arrangements of ideal type A ℐ . These stem from ideals ℐ in the set of positive roots of a reduced root system. We show that the K ( π , 1 ) –property holds for all arrangements A ℐ if the underlying Weyl group is classical and that it extends to most of the A ℐ if the underlying Weyl group is of exceptional type. Conjecturally this holds for all A ℐ . In general, the A ℐ are neither simplicial nor is their complexification of fiber type.

Freeness of multi-reflection arrangements via primitive vector fields

In 2002, Terao showed that every reflection multi-arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations. We first generalize Terao's result to multi-arrangements stemming from well-generated unitary reflection groups, where the multiplicity of a hyperplane depends on the order of its stabilizer. Here the exponents depend on the exponents of the dual reflection representation. We then extend our results further to all imprimitive irreducible unitary reflection groups. In this case the exponents turn out to depend on the exponents of a certain Galois twist of the dual reflection representation that comes from a Beynon-Lusztig type semi-palindromicity of the fake degrees.

Noncrossing partitions and Milnor fibers

For a finite real reflection group $W$ we use non-crossing partitions of type $W$ to construct finite cell complexes with the homotopy type of the Milnor fiber of the associated $W$-discriminant $\Delta_W$ and that of the Milnor fiber of the defining polynomial of the associated reflection arrangement. These complexes support natural cyclic group actions realizing the geometric monodromy. Using the shellability of the non-crossing partition lattice, this cell complex yields a chain complex of homology groups computing the integral homology of the Milnor fiber of $\Delta_W$.

Walls in Milnor Fiber Complexes

For a real reflection group the reflecting hyperplanes cut out on the unit sphere a simplicial complex called the Coxeter complex. Abramenko showed that each reflecting hyperplane meets the Coxeter complex in another Coxeter complex if and only if the Coxeter diagram contains no subdiagram of type D_4 , F_4 or H_4 . The present paper extends Abramenko's result to a wider class of complex reflection groups. These groups have a Coxeter-like presentation and a Coxeter-like complex called the Milnor fiber complex. Our first main theorem classifies the groups whose reflecting hyperplanes meet the Milnor fiber complex in another Milnor fiber complex. To understand better the walls that fail to be Milnor fiber complexes we introduce Milnor walls. Our second main theorem generalizes Abramenko's result in a second way. It says that each wall of a Milnor fiber complex is a Milnor wall if and only if the diagram contains no subdiagram of type D_4 , F_4 , or H_4 .

CLASSIFICATION OF REFLECTION SUBGROUPS MINIMALLY CONTAINING -SYLOW SUBGROUPS

Let a prime $p$ divide the order of a finite real reflection group. We classify the reflection subgroups up to conjugacy that are minimal with respect to inclusion, subject to containing a $p$-Sylow subgroup. For Weyl groups, this is achieved by an algorithm inspired by the Borel–de Siebenthal algorithm. The cases where there is not a unique conjugacy class of reflection subgroups minimally containing the $p$-Sylow subgroups are the groups of type $F_{4}$ when $p=2$ and $I_{2}(m)$ when $m\geq 6$ is even but not a power of $2$ for each odd prime divisor $p$ of $m$. The classification significantly reduces the cases required to describe the $p$-Sylow subgroups of finite real reflection groups.

Evidence for parking conjectures

Let W be an irreducible real reflection group. Armstrong, Reiner, and the author presented a model for parking functions attached to W[3] and made three increasingly strong conjectures about these objects. The author generalized these parking objects and conjectures to the Fuss–Catalan level of generality [26]. Even the weakest of these conjectures would uniformly imply a collection of facts in Coxeter–Catalan theory which are at present understood only in a case-by-case fashion. We prove that when W belongs to any of the infinite families ABCDI, the strongest of these conjectures is generically true.

Canonical Systems of Basic Invariants for Unitary Reflection Groups

AbstractIt is known that there exists a canonical system for every finite real reflection group. In a previous paper, the first and the third authors obtained an explicit formula for a canonical system. In this article, we first define canonical systems for the finite unitary reflection groups, and then prove their existence. Our proof does not depend on the classification of unitary reflection groups. Furthermore, we give an explicit formula for a canonical system for every unitary reflection group.

Spectra of Cayley graphs of complex reflection groups

Renteln proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. We prove the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provide a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups.

On Dunkl angular momenta algebra

We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincare-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl(N) version of the subalgebra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.