Root system chip-firing I: interval-firing

The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a 2004 paper. Let X be a finite set, C be a finite cyclic group acting on X, and f(q) be a polynomial in q with nonnegative integer coefficients. Then the triple (X,C,f(q)) exhibits the cyclic sieving phenomenon if, for all g in C, we have # X^g = f(w) where # denotes cardinality, X^g is the fixed point set of g, and w is a root of unity chosen to have the same order as g. It might seem improbable that substituting a root of unity into a polynomial with integer coefficients would have an enumerative meaning. But many instances of the cyclic sieving phenomenon have now been found. Furthermore, the proofs that this phenomenon hold often involve interesting and sometimes deep results from representation theory. We will survey the current literature on cyclic sieving, providing the necessary background about representations, Coxeter groups, and other algebraic aspects as needed.

The goals of this paper are two-fold. First, we prove, for an arbitrary finite root system D, the periodicity conjecture of Al. B. Zamolodchikov [24] that concerns Y-systems, a particular class of functional relations playing an important role in the theory of thermodynamic Bethe ansatz. Algebraically, Y-systems can be viewed as families of rational functions defined by certain birational recurrences formulated in terms of the root system (. We obtain explicit formulas for these rational functions, which always turn out to be Laurent polynomials, and prove that they exhibit the periodicity property conjectured by Zamolodchikov. In a closely related development, we introduce and study a simplicial complex A(b), which can be viewed as a generalization of the Stasheff polytope (also known as associahedron) for an arbitrary root system (D. In type A, this complex is the face complex of the ordinary associahedron, whereas in type B, our construction produces the Bott-Taubes polytope, or cyclohedron. We enumerate the faces of the complex A(@), prove that its geometric realization is always a sphere, and describe it in concrete combinatorial terms for the classical types ABCD. The primary motivation for this investigation came from the theory of cluster algebras, introduced in [9] as a device for studying dual canonical bases and total positivity in semisimple Lie groups. This connection remains behind the scenes in the text of this paper, and will be brought to light in a forthcoming sequel1 to [9].

Jack polynomials and free cumulants

We investigate the uniqueness of factorisation of possibly disconnected finite graphs with respect to the Cartesian, the strong and the direct product. It is proved that if a graph has n connected components, where n is prime, or n = 1, 4, 8, 9, and satisfies some additional natural conditions, it factors uniquely under the given products. If, on the contrary, n = 6 or 10, all cases of nonunique factorisation are described precisely.

Gessel conjectured that the two-sided Eulerian polynomial, recording the common distribution of the descent number of a permutation and that of its inverse, has non-negative integer coefficients when expanded in terms of the gamma basis. This conjecture has been proved recently by Lin.We conjecture that an analogous statement holds for simple permutations, and use the substitution decomposition tree of a permutation (by repeated inflation) to show that this would imply the Gessel-Lin result. We provide supporting evidence for this stronger conjecture.

An alternate derivation of the dual condition (called the severance-value condition in this paper) to feasibility of the multicommodity flow problem is given by graph theoretical arguments. That is, a multicommodity flow of given requirements is feasible if and only if the capacity of every severance is no less than its least capacity consumption for the flow. A severance is a set of the edges with nonnegative integer coefficients. Even when the severance-value condition is satisfied by a certain finite subset of severances, the multicommodity flow is shown to be still feasible if each requirement is allowed to be reduced by an appropriate amount \mu . This truncation allowance \mu is estimated in terms of the network topology and the capacity function.

A combinatorial interpretation for two transformations of series that commute with compositional inversion

The Kashiwara \(B(\infty )\) crystal pertains to a Verma module for a Kac–Moody algebra. It has a rich and important combinatorial structure. A Preparation Theorem is established as a preliminary step in an explicit description \(B(\infty )\). It is noted that \(B(\infty )\) has in general infinitely many presentations as subsets of countably many copies of the natural numbers each given by successive reduced decompositions of Weyl group elements. In each presentation there is an action of Kashiwara operators determined by Kashiwara functions. These functions are linear in the entries. Thus a natural question is to show that in each presentation the subset \(B(\infty )\) is polyhedral and to describe explicitly the linear inequalities involved. Here an approach to this question is initiated based on constructing dual Kashiwara functions. Our basic hypothesis is that dual Kashiwara functions exist and can be expressed as differences of successive Kashiwara functions with non-negative integer coefficients. This last requirement is called the positivity condition. Here one starts from an explicit linear function called the “initial driving function”. Then in terms of the reduced decomposition one seeks an algorithm to provide further dual Kashiwara functions which in particular will provide “driving functions” for the next induction step. The Preparation Theorem resolves a main difficulty in this step-wise construction, namely that the resulting functions must exhibit an invariance with respect to the Kashiwara operators. This is expressed through a sum, or simply S, condition. It depends very subtly on inequalities between the coefficients occurring in the driving function obtained from the previous step. The proof of the Preparation Theorem depends on the construction of S-graphs constructed by a process of “binary fusion” based on the structure of the driving function.

In a previous work, exact formulae and differential equations were found for traces of powers of the zero mode in the W3 algebra. In this paper we investigate their modular properties, in particular we find the exact result for the modular transformations of traces of $W_0^n$ for n = 1, 2, 3, solving exactly the problem studied approximately by Gaberdiel, Hartman and Jin. We also find modular differential equations satisfied by traces with a single $W_0$ inserted, and relate them to differential equations studied by Mathur et al. We find that, remarkably, these all seem to be related to weight 0 modular forms with expansions with non-negative integer coefficients.

Uniform lower bound for the least common multiple of a polynomial sequence

formula omitted]-partition power sums

We construct “quantum theta bases,” extending the set of quantum cluster monomials, for various versions of skew-symmetric quantum cluster algebras. These bases consist precisely of the indecomposable universally positive elements of the algebras they generate, and the structure constants for their multiplication are Laurent polynomials in the quantum parameter with non-negative integer coefficients, proving the quantum strong cluster positivity conjecture for these algebras. The classical limits recover the theta bases considered by Gross–Hacking–Keel–Kontsevich (J Am Math Soc 31(2):497–608, 2018). Our approach combines the scattering diagram techniques used in loc. cit. with the Donaldson–Thomas theory of quivers.

We introduce and study a family of simplicial complexes associated to an arbitrary finite root system and a nonnegative integer parameter m. For m=1, our construction specializes to the (simplicial) generalized associahedra or, equivalently, to the cluster complexes for the cluster algebras of finite type. Our computation of the face numbers and h-vectors of these complexes produces the enumerative invariants defined in other contexts by C.A.Athanasiadis, suggesting links to a host of well studied problems in algebraic combinatorics of finite Coxeter groups, root systems, and hyperplane arrangements. Recurrences satisfied by the face numbers of our complexes lead to combinatorial algorithms for determining Coxeter-theoretic invariants. That is, starting with a Coxeter diagram of a finite Coxeter group, one can compute the Coxeter number, the exponents, and other classical invariants by a recursive procedure that only uses most basic graph-theoretic concepts applied to the input diagram. In types A and B, we rediscover the constructions and results obtained by E.Tzanaki .

For a given real polynomial f without positive roots (without nonnegative roots, respectively) we study polynomials h of lowest degree such that the product hf has nonnegative (positive, respectively) coefficients. Our main result is the determination of polynomials h of minimal degree with the aforementioned properties for quadratic polynomials f with negative discriminant; here we exploit slightly modified ideas of E. Meissner (1911). Further, we show that h can be calculated in finitely many steps.

In a previous paper, we introduced functions $\zeta_r(s,a,\Delta;q)$, which are $q$-analogues of zeta functions associated with root systems $\Delta$, and their $p$-deformations $\zeta_r(s,a,\beta,\Delta;p,q)$. It is known that zeta functions of root systems satisfy certain functional relations including Witten's volume formula. In this paper, we consider $q$-extensions of these functional relations for $\Delta=\Delta(A_2), \ \Delta(A_3), \ \Delta(B_2), \ \Delta(G_2)$. We also generalize expressions of ``Weyl group symmetric'' linear combination of $\zeta_r(s,a,\beta,\Delta;p,q)$ for $\Delta=\Delta(A_1), \ \Delta(A_2), \ \Delta(A_3)$, which were obtained in the previous paper, to arbitrary root systems $\Delta$ by using an identity due to Macdonald.