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We prove that the centralizer of a Coxeter element in an irreducible Coxeter group is the cyclic group generated by that Coxeter element. [ABSTRACT FROM AUTHOR]

The characterization of orbits of roots under the action of a Coxeter element is a fundamental tool in the study of finite root systems and their reflection groups. This paper develops the analogous tool in the affine setting, adding detail and uniformity to a result of Dlab and Ringel. [ABSTRACT FROM AUTHOR]

Interchanging the character and co-character groups of a torus T over a field k introduces a contravariant functor T → T∨. Interpreting ρ : T(ℂ) → ℂ×, half the sum of positive roots for T, a maximal torus in a simply connected semi-simple group G (over ℂ) using this duality, we get a co-character ρ∨ : ℂ× → T∨(ℂ) for which ρ∨( e2π i/ h) ( h the Coxeter number) is the Coxeter conjugacy class of the dual group G∨(ℂ). This point of view gives a rather transparent proof of a theorem of Kostant on the character values of irreducible finite-dimensional representations of G(ℂ) at the Coxeter conjugacy class: the proof amounting to the fact that in G∨sc(ℂ), the simply connected cover of G∨(ℂ), there is a unique regular conjugacy class whose image in G∨(ℂ) has order h (which is the Coxeter conjugacy class). [ABSTRACT FROM AUTHOR]

Chapuy and Stump have given a nice generating series for the number of factorizations of a Coxeter element as a product of reflections. Their method is to evaluate case by case a character-theoretic expression. The goal of this note is to give a uniform evaluation of their character-theoretic expression in the case of Weyl groups, by using combinatorial properties of Deligne-Lusztig representations. [ABSTRACT FROM AUTHOR]

Let $${\mathfrak {g}}$$ be a simple Lie algebra of rank r over $$\mathbb {C}, {\mathfrak {h}}\subset {\mathfrak {g}}$$ a Cartan subalgebra. We construct a family of r commuting Hermitian operators acting on $${\mathfrak {h}}$$ whose eigenvalues are equal to the coordinates of the eigenvectors of the Cartan matrix of $${\mathfrak {g}}$$ . [ABSTRACT FROM AUTHOR]

Abstract: Let be a complex simple Lie algebra and a Borel subalgebra. The algebra Y of polynomial semi-invariants on the dual of is a polynomial algebra on rank generators (Grothendieck and Dieudonné (1965–1967)) . The analogy with the semisimple case suggests there exists an algebraic slice to coadjoint action, that is an affine translate of a vector subspace of such that the restriction map induces an isomorphism of Y onto the algebra of regular functions on . This holds in type A and even extends to all biparabolic subalgebras (Joseph (2007)) ; but the construction fails in general even with respect to the Borel. Moreover already in type no algebraic slice exists. Very surprisingly the exception of type is itself an exception. Indeed an algebraic slice for the coadjoint action of the Borel subalgebra is constructed for all simple Lie algebras except those of types , and . Outside type A, the slice obtained meets an open dense subset of regular orbits, even though the special point y of the slice is not itself regular. This explains the failure of our previous construction. [Copyright &y& Elsevier]

Let N ⊴ G be a pair of finite subgroups of SL2(ℂ) and V a finite-dimensional fundamental G-module. We study Kostant’s generating functions for the decomposition of the SL2(ℂ)-module Sk(V ) restricted to N ◃ G in connection with the McKay–Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincaré series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined. [ABSTRACT FROM AUTHOR]

We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of finite "vertically and laterally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type). Furthermore, we show that every picture for such a set of positive roots is a linear combination of "atoms" and we give a precise description of atoms as special semi-invariant pictures. [ABSTRACT FROM AUTHOR]

Although both noncrossing partitions and nonnesting partitions are uniformly enumerated for Weyl groups, the exact relationship between these two sets of combinatorial objects remains frustratingly mysterious. In this paper, we give a precise combinatorial answer in the case of the symmetric group: for any standard Coxeter element, we construct an equivariant bijection between noncrossing partitions under the Kreweras complement and nonnesting partitions under a Coxeter‐theoretically natural cyclic action we call the Kroweras complement. Our equivariant bijection is the unique bijection that is both equivariant and support‐preserving, and is built using local rules depending on a new definition of charmed roots. Charmed roots are determined by the choice of Coxeter element — in the special case of the linear Coxeter element (1,2,...,n)$(1,2,\ldots ,n)$, we recover one of the standard bijections between noncrossing and nonnesting partitions. [ABSTRACT FROM AUTHOR]

Let G be a semi-simple simply connected algebraic group over the field ℂ of complex numbers. Let T be a maximal torus of G, and let W be the Weyl group of G with respect to T. Let Z(w, i) be the Bott-Samelson-Demazure-Hansen variety corresponding to a tuple i associated to a reduced expression of an element w ∈ W. We prove that for the tuple i associated to any reduced expression of a minuscule Weyl group element w, the anti-canonical line bundle on Z(w, i) is globally generated. As consequence, we prove that Z(w, i) is weak Fano. Assume that G is a simple algebraic group whose type is different from A2. Let S = {α1, ..., αn} be the set of simple roots. Let w be such that support of w is equal to S. We prove that Z(w, i) is Fano for the tuple i associated to any reduced expression of w if and only if w is a Coxeter element and w − 1 (∑ t = 1 n α t) ∈ − S . [ABSTRACT FROM AUTHOR]

The ıquantum$\imath {\rm quantum}$ group Unı${\mathrm{U}^\imath _{n}}$ of type AIn$\textrm {AI}_n$ is a coideal subalgebra of the quantum group Uq(sln+1)$U_q(\mathfrak {sl}_{n+1})$, associated with the symmetric pair (sln+1,son+1)$(\mathfrak {sl}_{n+1},\mathfrak {so}_{n+1})$. In this paper, we give a cluster realisation of the algebra Unı${\mathrm{U}^\imath _{n}}$. Under such a realisation, we give cluster interpretations of some fundamental constructions of Unı${\mathrm{U}^\imath _{n}}$, including braid group symmetries, the coideal structure and the action of a Coxeter element. Along the way, we study a (rescaled) integral form of Unı${\mathrm{U}^\imath _{n}}$, which is compatible with our cluster realisation. We show that this integral form is invariant under braid group symmetries, and construct the Poincare‐Birkhoff‐Witt (PBW)‐bases for the integral form. [ABSTRACT FROM AUTHOR]

Abstract: Let be an affine Weyl group and be the natural projection to the corresponding finite Weyl group. We say that has finite Coxeter part if is conjugate to a Coxeter element of . The elements with finite Coxeter part are a union of conjugacy classes of . We show that for each conjugacy class of with finite Coxeter part there exists a unique maximal proper parabolic subgroup of , such that the set of minimal length elements in is exactly the set of Coxeter elements in . Similar results hold for twisted conjugacy classes. [Copyright &y& Elsevier]

Abstract: In this paper we show that for a simply-laced root system a choice of a Coxeter element C gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to C-orbits in R; moreover, it gives an identification of R with a certain subset of , where h is the Coxeter number. The set has a natural quiver structure; we call it the periodic Auslander–Reiten quiver. This gives a combinatorial construction of the root system associated with the Dynkin diagram I: roots are vertices of , and the root lattice and the inner product admit an explicit description in terms of . Finally, we relate this construction to the theory of quiver representations. [Copyright &y& Elsevier]

We initiate the study of a class of polytopes, which we coin polypositroids , defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and alcoved polytopes. Whereas positroids are the matroids arising from the totally nonnegative Grassmannian, polypositroids are "positive" polymatroids. We parametrize polypositroids using Coxeter necklaces and balanced graphs, and describe the cone of polypositroids by extremal rays and facet inequalities. We introduce a notion of $(W,c)$ -polypositroid for a finite Weyl group W and a choice of Coxeter element c. We connect the theory of $(W,c)$ -polypositroids to cluster algebras of finite type and to generalized associahedra. We discuss membranes , which are certain triangulated 2-dimensional surfaces inside polypositroids. Membranes extend the notion of plabic graphs from positroids to polypositroids. [ABSTRACT FROM AUTHOR]

We consider presentations that were derived in [3] for the interval groups associated with proper quasi-Coxeter elements of the Coxeter group W (D n). We use combinatorial methods to derive alternative presentations for the groups, and use these new presentations to show that the interval group associated with a proper quasi-Coxeter element of W (D n) cannot be isomorphic to the Artin group of type D n. While the specific problems we solve arise from the study of interval groups, their solution provides an illustration of how techniques indicated by computational observation can be used to derive properties of all groups within an infinite family. [ABSTRACT FROM AUTHOR]

If f denotes the truncated Lusztig Fourier transform, we show that the image by f of the normalized characteristic function of a Coxeter element is the alternate sum of the exterior powers of the reflection representation, and that any class function is tower equivalent to its image by f. In particular this gives a proof of the results of Chapuy and Douvropoulos on "Coxeter factorizations with generalized Jucys-Murphy weights and matrix tree theorems for reflection groups" for irreducible spetsial reflection groups, based on Deligne-Lusztig combinatorics. [ABSTRACT FROM AUTHOR]

Abstract: We consider Fuchsian singularities of arbitrary genus and prove, in a conceptual manner, a formula for their Poincaré series. This uses Coxeter elements involving Eichler–Siegel transformations. We give geometrical interpretations for the lattices and isometries involved, lifting them to triangulated categories. [ABSTRACT FROM AUTHOR]

In this paper, for any simple, simply connected algebraic group G of type B,C or D and for any maximal parabolic subgroup P of G, we describe all minimal dimensional Schubert varieties in G/P admitting semistable points for the action of a maximal torus T with respect to an ample line bundle on G/P. We also describe, for any semi-simple simply connected algebraic group G and for any Borel subgroup B of G, all Coxeter elements τ for which the Schubert variety X(τ) admits a semistable point for the action of the torus T with respect to a non-trivial line bundle on G/B. [ABSTRACT FROM AUTHOR]

Several combinatorial actions of the affine Weyl group of type C ˜ n on triangulations, trees, words and permutations are compared. Addressing a question of David Vogan, we show that, modulo a natural involution, these permutation representations are multiplicity-free. The proof uses a general construction of Gelfand subgroups in the affine Weyl groups of types C ˜ n and B ˜ n. [ABSTRACT FROM AUTHOR]

For a finite irreducible Coxeter group (W , S) with a fixed Coxeter element c and set of reflections T , Defant and Williams define a pop-tsack torsing operation Po p T : W → W given by Po p T (w) = w ⋅ π (w) − 1 where π (w) = ⋁ t ≤ T w , t ∈ T N C (w , c) t is the join of all reflections lying below w in the absolute order in the non-crossing partition lattice N C (w , c). This is a "dual" notion of the pop-stack sorting operator Po p S ; Po p S was introduced by Defant as a way to generalize the pop-stack sorting operator on S n to general Coxeter groups. Define the forward orbit of an element w ∈ W to be O Po p T (w) = { w , Po p T (w) , Po p T 2 (w) , ... }. Defant and Williams established the length of the longest possible forward orbits max w ∈ W ⁡ | O Po p T (w) | for Coxeter groups of coincidental types and Type D in terms of the corresponding Coxeter number of the group. In their paper, they also proposed multiple conjectures about enumerating elements with near maximal orbit length. We resolve all the conjectures that they have put forth about enumeration, and in the process we give complete classifications of these elements of Coxeter groups of types A, B and D with near maximal orbit lengths. [ABSTRACT FROM AUTHOR]

We introduce a new family of finite posets which we call 2 -chains. These first arose in the study of 0-Hecke algebras, but they admit a variety of different characterisations. We give these characterisations, prove that they are equivalent and derive some numerical results concerning 2 -chains. [ABSTRACT FROM AUTHOR]

Inspired by the quantum McKay correspondence, we consider the classical ADE Lie theory as a quantum theory over sl 2 . We introduce anti-symmetric characters for representations of quantum groups and investigate the Fourier duality to study the spectral theory. In the ADE Lie theory, there is a correspondence between the eigenvalues of the Coxeter element and the eigenvalues of the adjacency matrix. We formalize related notions and prove such a correspondence for representations of Verlinde algebras of quantum groups: this includes generalized Dynkin diagrams over any simple Lie algebra g at any level k . This answers a recent comment of Terry Gannon on an old question posed by Victor Kac in 1994. [ABSTRACT FROM AUTHOR]

Let G be a simple algebraic group of adjoint type over C , whose root system is of type F 4. Let T be a maximal torus of G and B be a Borel subgroup of G containing T. Let w be an element of the Weyl group W and X(w) be the Schubert variety in the flag variety G/B corresponding to w. Let Z (w , i ̲) be the Bott–Samelson–Demazure–Hansen variety (the desingularization of X(w)) corresponding to a reduced expression i ̲ of w. In this article, we study the cohomology modules of the tangent bundle on Z (w 0 , i ̲) , where w 0 is the longest element of the Weyl group W. We describe all the reduced expressions of w 0 in terms of a Coxeter element such that Z (w 0 , i ̲) is rigid (see Theorem 7.1). Further, if G is of type G 2 , there is no reduced expression i ̲ of w 0 for which Z (w 0 , i ̲) is rigid (see Theorem 8.2). [ABSTRACT FROM AUTHOR]

The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac–Moody group—the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang–Zelevinsky in finite type. In type |$A_{n}^{\!(1)}$| and finitely many other affine types, we show that cluster variables of regular quiver representations are realized by generalized minors of group representations that are neither highest- nor lowest-weight; we conjecture this holds more generally. [ABSTRACT FROM AUTHOR]

Let G be a simply connected simple algebraic group over C , B and B − be its two opposite Borel subgroups. For two elements u , v of the Weyl group W , it is known that the coordinate ring C [ G u , v ] of the double Bruhat cell G u , v = B u B ∩ B − v B − is isomorphic to a cluster algebra A (i) C [2,12]. In the case u = e , v = c 2 (c is a Coxeter element), the algebra C [ G e , c 2 ] has only finitely many cluster variables. In this article, for G = SL r + 1 (C) , we obtain explicit forms of all the cluster variables in C [ G e , c 2 ] by considering its additive categorification via preprojective algebras, and describe them in terms of monomial realizations of Demazure crystals. [ABSTRACT FROM AUTHOR]

Let G ˜ = GL (n , F ¯ q) where q is a power of a prime. Let F be the standard Frobenius map of G ˜. Let B ˜ denote the Borel subgroup of G ˜ of upper triangular matrices in G ˜ and let T ˜ denote the maximal split torus of G ˜ contained in B ˜. Let W denote the Weyl group of the B ˜ , N ˜ pair where N ˜ = N G ˜ (T ˜). Let w ∈ W and let w ˙ denote an element in the pre-image of the natural surjection map from N G ˜ (T ˜) to W. The double coset B ˜ w ˙ B ˜ contains only regular elements if and only if w is a Coxeter element of minimal length. In the following we consider an element w ∈ W that is not of minimal length and thus B ˜ w B ˜ contains nonregular elements. We explicitly describe certain cosets that are subsets of B ˜ w B ˜ and contain only regular elements. These cosets are chosen for their importance in the construction of a basis of a Hecke algebra associated with G = G ˜ F. [ABSTRACT FROM AUTHOR]

Abstract For a classical group G and a Coxeter element c of the Weyl group, it is known that the coordinate ring ℂ [ G e , c 2 ] of the double Bruhat cell G e , c 2 ≔ B ∩ B − c 2 B − has a structure of cluster algebra of finite type, where B and B − are opposite Borel subgroups. In this article, we consider the case G is of type B r , C r or D r and describe all the cluster variables in ℂ [ G e , c 2 ] as monomial realizations of certain Demazure crystals. [ABSTRACT FROM AUTHOR]

recent experimental discovery involving the spin structure of electrons in a cold one-dimensional magnet points to a validation of a (1989) Zamolodchikov model involving the exceptional Lie group E. The model predicts 8 particles and predicts the ratio of their masses. The conjectures have now been validated experimentally, at least for the first five masses. The Zamolodchikov model was extended in 1990 to a Kateev–Zamolodchikov model involving E and E as well. In a seemingly unrelated matter, the vertices of the 8-dimensional Gosset polytope identifies with the 240 roots of E. Under the famous two-dimensional (Peter McMullen) projection of the polytope, the images of the vertices are arranged in eight concentric circles, hereafter referred to as the Gosset circles. The McMullen projection generalizes to any complex simple Lie algebra (in particular not restricted to A-D-E types) whose rank is greater than 1. The Gosset circles generalize as well, using orbits of the Coxeter element on roots. Applying results in Kostant (Am J Math 81:973–1032, 1959), I found some time ago a very easily defined operator A on a Cartan subalgebra, the ratio of whose eigenvalues is exactly the ratio of squares of the radii r of the generalized Gosset circles. The two matters considered above relate to one another in that the ratio of the masses in the E, E, E Kateev–Zamolodchikov models are exactly equal to the ratios of the radii of the corresponding generalized Gosset circles. [ABSTRACT FROM AUTHOR]

In contrast to the classical and semiclassical settings, the Coxeter element $ (12\dots n)$ matrix does not determine an automorphism of the quantum grassmannian. Here, we show that this cycling can be obtained by means of a cocycle twist. A consequence is that the torus invariant prime ideals of the quantum grassmannian are permuted by the action of the Coxeter element $ (12\dots n)$ $ (12\dots n)$ [ABSTRACT FROM AUTHOR]

Abstract: We study the Hurwitz action of the classical braid group on factorisations of a Coxeter element c in a well-generated complex reflection group W. It is well known that the Hurwitz action is transitive on the set of reduced decompositions of c in reflections. Our main result is a similar property for the primitive factorisations of c, i.e. factorisations with only one factor which is not a reflection. The motivation is the search for a geometric proof of Chapoton''s formula for the number of chains of given length in the non-crossing partitions lattice . Our proof uses the properties of the Lyashko–Looijenga covering and the geometry of the discriminant of W. [Copyright &y& Elsevier]

Given a finite irreducible Coxeter group $W$, a positive integer $d$, and types $T_1,T_2,dots ,T_d$ (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions $c=sigma _1sigma _2cdots sigma _d$ of a Coxeter element $c$ of $W$, such that $sigma _i$ is a Coxeter element in a subgroup of type $T_i$ in $W$, $i=1,2,dots ,d$, and such that the factorisation is ``minimal'''' in the sense that the sum of the ranks of the $T_i$''s, $i=1,2,dots ,d$, equals the rank of $W$. For the exceptional types, these decomposition numbers have been computed by the first author in [{it ``Topics in Discrete Mathematics,''''} M. Klazar et al. (eds.), Springer--Verlag, Berlin, New York, 2006, pp. 93--126] and [{it Séminaire Lotharingien Combin.} {bf 54} (2006), Article B54l]. The type $A_n$ decomposition numbers have been computed by Goulden and Jackson in [{it Europ. J. Combin.} {bf 13} (1992), 357--365], albeit using a somewhat different language. We explain how to extract the type $B_n$ decomposition numbers from results of Bóna, Bousquet, Labelle and Leroux [Adv. Appl. Math. {bf 24} (2000), 22--56] on map enumeration. Our formula for the type $D_n$ decomposition numbers is new. These results are then used to determine, for a fixed positive integer $l$ and fixed integers $r_1le r_2le dots le r_l$, the number of multi-chains $pi _1le pi _2le dots le pi _l$ in Armstrong''s generalised non-crossing partitions poset, where the poset rank of $pi _i$ equals $r_i$ and where the ``block structure'''' of $pi _1$ is prescribed. We demonstrate that this result implies all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations. Surprisingly, this result on multi-chain enumeration is new even for the original non-crossing partitions of Kreweras. Moreover, the result allows one to solve the problem of rank-selected chain enumeration in the type $D_n$ generalised non-crossing partitions poset, which, in turn, leads to a proof of Armstrong''s $F=M$ Conjecture in type $D_n$, thus completing a computational proof of the $F=M$ Conjecture for all types. It also allows one to address another conjecture of Armstrong on maximal intervals containing a random multi-chain in the generalised non-crossing partitions poset. [ABSTRACT FROM AUTHOR]

In this paper we introduce abstract string modules and give an explicit bijection between the submodule lattice of an abstract string module and the perfect matching lattice of the corresponding abstract snake graph. In particular, we make explicit the direct correspondence between a submodule of a string module and the perfect matching of the corresponding snake graph. For every string module we define a Coxeter element in a symmetric group. We then establish a bijection between the submodule lattice of the string module and the lattice given by the interval in the weak Bruhat order determined by the Coxeter element. Using the correspondence between string modules and snake graphs, we give a new concise formulation of snake graph calculus. [ABSTRACT FROM AUTHOR]

We call an element of a Coxeter group a parabolic quasi-Coxeter element if it has a reduced decomposition into a product of reflections that generate a parabolic subgroup. We show that for a parabolic quasi-Coxeter element in an affine Coxeter group the Hurwitz action on its set of reduced decompositions into a product of reflections is transitive. [ABSTRACT FROM AUTHOR]

We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element.We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a factorization into a product of reflections that generate a parabolic subgroup. We give an unusual definition of a parabolic subgroup that we show to be equivalent to the classical one for finite Coxeter groups. [ABSTRACT FROM AUTHOR]

We present a categorification of the non-crossing partitions given by crystallographic Coxeter groups. This involves a category of certain bilinear lattices, which are essentially determined by a symmetrisable generalised Cartan matrix together with a particular choice of a Coxeter element. Examples arise from Grothendieck groups of hereditary artin algebras. [ABSTRACT FROM AUTHOR]

We prove that the restriction of Bruhat order to type A noncrossing partitions for the Coxeter element c = s 1 s 2 ⋯ s n defines a distributive lattice isomorphic to the order ideals of the root poset ordered by inclusion. Motivated by the base change from the graphical basis of the Temperley–Lieb algebra to the image of the simple elements of the dual braid monoid, we extend this bijection to other Coxeter elements using certain canonical factorizations. In particular, we introduce a new set of vectors counted by the Catalan numbers and give new bijections–fixing each reflection–between noncrossing partitions associated to distinct Coxeter elements. [ABSTRACT FROM AUTHOR]

We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. We derive toric analogues for several features of ordinary partial orders, such as chains, antichains, transitivity, Hasse diagrams, linear extensions, and total orders. [ABSTRACT FROM AUTHOR]

In this article, we give a short algebraic proof that all closed intervals in a γ -Cambrian semilattice C γ are trim for any Coxeter group W and any Coxeter element γ ∈ W . This means that if such an interval has length k , then there exists a maximal chain of length k consisting of left-modular elements, and there are precisely k join- and k meet-irreducible elements in this interval. Consequently, every graded interval in C γ is distributive. This problem was open for any Coxeter group that is not a Weyl group. [ABSTRACT FROM AUTHOR]

Let Q be an acyclic quiver and s be a sequence with elements in the vertex set Q0. We describe an induced sequence of simple (backward) tilting in the bounded derived category D(Q), starting from the standard heart HQ = modkQ and ending at another heart Hs in D(Q). Then we show that s is a green mutation sequence if and only if every heart in this simple tilting sequence is greater than or equal to HQ[-1]; it is maximal if and only if Hs = HQ[-1]. This provides a categorical way to understand green mutations. Further, fix a Coxeter element c in the Coxeter group WQ of Q, which is admissible with respect to the orientation of Q. We prove that the sequence w induced by a c-sortable word w is a green mutation sequence. As a consequence, we obtain a bijection between c-sortable words and finite torsion classes in HQ. As byproducts, the interpretations of inversions, descents and cover reflections of a c-sortable word w are given in terms of the combinatorics of green mutations. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to introduce a generalized quantum cluster character, which assigns to each object $$V$$ of a finitary Abelian category $${\mathcal {C}}$$ over a finite field $${\mathbb {F}}_q$$ and any sequence $$\mathbf {i}$$ of simple objects in $${\mathcal {C}}$$ the element $$X_{V,\mathbf {i}}$$ of the corresponding algebra $$P_{{\mathcal {C}},\mathbf {i}}$$ of $$q$$ -polynomials. We prove that if $${\mathcal {C}}$$ was hereditary, then the assignments $$V\mapsto X_{V,\mathbf {i}}$$ define algebra homomorphisms from the (dual) Hall-Ringel algebra of $${\mathcal {C}}$$ to the $$P_{{\mathcal {C}},\mathbf {i}}$$ , which generalize the well-known Feigin homomorphisms from the upper half of a quantum group to $$q$$ -polynomial algebras. If $${\mathcal {C}}$$ is the representation category of an acyclic valued quiver $$(Q,\mathbf {d})$$ and $$\mathbf {i}=(\mathbf {i}_0,\mathbf {i}_0)$$ , where $$\mathbf {i}_0$$ is a repetition-free source-adapted sequence, then we prove that the $$\mathbf {i}$$ -character $$X_{V,\mathbf {i}}$$ equals the quantum cluster character $$X_V$$ introduced earlier by the second author in Rupel (Int Math Res Not 14:3207-3236, ; Quantum cluster characters of valued quivers, ). Using this identification, we deduce a quantum cluster structure on the quantum unipotent cell corresponding to the square of a Coxeter element. As a corollary, we prove a conjecture from the joint paper Berenstein and Zelevinsky (Adv Math 195(2):405-455, ) of the first author with A. Zelevinsky for such quantum unipotent cells. As a by-product, we construct the quantum twist and prove that it preserves the triangular basis introduced by A. Zelevinsky and the first author in Berenstein and Zelevinsky (Int Math Res Not. doi:, ). [ABSTRACT FROM AUTHOR]

Let W be a Weyl group with root lattice Q and Coxeter number h . The elements of the finite torus Q / ( h + 1 ) Q are called the W - parking functions , and we call the permutation representation of W on the set of W -parking functions the (standard) W - parking space . Parking spaces have interesting connections to enumerative combinatorics, diagonal harmonics, and rational Cherednik algebras. In this paper we define two new W -parking spaces, called the noncrossing parking space and the algebraic parking space , with the following features: • They are defined more generally for real reflection groups. • They carry not just W -actions, but W × C -actions, where C is the cyclic subgroup of W generated by a Coxeter element. • In the crystallographic case, both are isomorphic to the standard W -parking space. Our Main Conjecture is that the two new parking spaces are isomorphic to each other as permutation representations of W × C . This conjecture ties together several threads in the Catalan combinatorics of finite reflection groups. Even the weakest form of the Main Conjecture has interesting combinatorial consequences, and this weak form is proven in all types except E 7 and E 8 . We provide evidence for the stronger forms of the conjecture, including proofs in some cases, and suggest further directions for the theory. [ABSTRACT FROM AUTHOR]

In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W also lie in this parabolic subgroup. [ABSTRACT FROM AUTHOR]

One possible way to obtain the quasicrystallographic structure is the projection of the higher-dimensional lattice into two- or three-dimensional subspaces. Here a general technique applicable to any higher-dimensional lattice is introduced. The Coxeter number and the integers of the Coxeter exponents of a Coxeter-Weyl group play a crucial role in determining the plane onto which the lattice is to be projected. The quasicrystal structures display the dihedral symmetry of order twice that of the Coxeter number. The eigenvectors and the corresponding eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors in n-dimensional Euclidean space which lead to suitable choices for the projection subspaces. The maximal dihedral subgroup of the Coxeter-Weyl group is identified to determine the symmetry of the quasicrystal structure. Examples are given for 12-fold symmetric quasicrystal structures obtained by projecting the higher-dimensional lattices determined by the affine Coxeter-Weyl groups W a( F4), W a( B6) and W a( E6). These groups share the same Coxeter number h = 12 with different Coxeter exponents. The dihedral subgroup D12 of the Coxeter groups can be obtained by defining two generators R1 and R2 as the products of generators of the Coxeter-Weyl groups. The reflection generators R1 and R2 operate in the Coxeter planes where the Coxeter element R1 R2 of the Coxeter-Weyl group represents the rotation of order 12. The canonical (strip, equivalently, cut-and-project technique) projections of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with fourfold and sixfold symmetry. It is noted that the quasicrystal structures obtained from the lattices W a( F4) and W a( B6) are compatible with some experimental results. [ABSTRACT FROM AUTHOR]

Let Q be an acyclic quiver and Λ be the complete preprojective algebra of Q over an algebraically closed field k. To any element w in the Coxeter group of Q, Buan, Iyama, Reiten and Scott [‘Cluster structures for 2-Calabi–Yau categories and unipotent groups’, Compos. Math. 145 (2009) 1035–1079] have introduced and studied a finite-dimensional algebra Λw=Λ/Iw. In this paper, we look at filtrations of Λw associated to any reduced expression w of w. We are especially interested in the case where the word w is c-sortable, where c is a Coxeter element. In this situation, the consecutive quotients of this filtration can be related to tilting kQ-modules with finite torsionfree class. [ABSTRACT FROM AUTHOR]

Abstract: Let be a Lie algebra of type A, D, E with fixed Cartan subalgebra , root system R and Weyl group W. We show that a choice of Coxeter element gives a root basis for . Moreover, using the results of Kirillov and Thind (2010) we show that this root basis gives a purely combinatorial construction of , where root vectors correspond to vertices of a certain quiver , and with respect to this basis the structure constants of the Lie bracket are given by paths in . This construction is then related to the constructions of Ringel and Peng and Xiao. [Copyright &y& Elsevier]

Abstract: Given a finite Coxeter system and a Coxeter element c, or equivalently an orientation of the Coxeter graph of W, we construct a simple polytope whose outer normal fan is N. Reading''s Cambrian fan , settling a conjecture of Reading that this is possible. We call this polytope the c-generalized associahedron. Our approach generalizes Loday''s realization of the associahedron (a type A c-generalized associahedron whose outer normal fan is not the cluster fan but a coarsening of the Coxeter fan arising from the Tamari lattice) to any finite Coxeter group. A crucial role in the construction is played by the c-singleton cones, the cones in the c-Cambrian fan which consist of a single maximal cone from the Coxeter fan. Moreover, if W is a Weyl group and the vertices of the permutahedron are chosen in a lattice associated to W, then we show that our realizations have integer coordinates in this lattice. [Copyright &y& Elsevier]

Abstract: We show that exceptional sequences for hereditary algebras are characterized by the fact that the product of the corresponding reflections is the inverse Coxeter element in the Weyl group. We use this result to give a new combinatorial characterization of clusters tilting sets in the cluster category in the case where the hereditary algebra is of finite type. [Copyright &y& Elsevier]

Abstract: For a simple complex Lie group G the connected components of the moduli space of semistable G-bundles over an elliptic curve are weighted projective spaces or quotients of weighted projective spaces by a finite group action. In this note we will provide a new proof of this result using the invariant theory of affine Kac–Moody groups, in particular the action of the (twisted) Coxeter element on the root system of G. [Copyright &y& Elsevier]

For a finite Coxeter group W and a Coxeter element c of W, the c-Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the c-sortable elements of W. The main result of this paper is that the known bijection clc between c-sortable elements and c-clusters induces a combinatorial isomorphism of fans. In particular, the c-Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for W. The rays of the c-Cambrian fan are generated by certain vectors in the W-orbit of the fundamental weights, while the rays of the c-cluster fan are generated by certain roots. For particular ("bipartite") choices of c, we show that the c-Cambrian fan is linearly isomorphic to the c-cluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map clc, on c-clusters by the c-Cambrian lattice. We give a simple bijection from c-clusters to c-noncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well-known objects in the theory of cluster algebras, providing a geometric context for g-vectors and quasi-Cartan companions. [ABSTRACT FROM AUTHOR]

Abstract: We construct a full strongly exceptional collection in the triangulated category of graded matrix factorizations of a polynomial associated to a nondegenerate regular system of weights whose smallest exponents are equal to −1. In the associated Grothendieck group, the strongly exceptional collection defines a root basis of a generalized root system of sign and a Coxeter element of finite order, whose primitive eigenvector is a regular element in the expanded symmetric domain of type IV with respect to the Weyl group. [Copyright &y& Elsevier]