Y -systems and generalized associahedra

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 .

Cluster algebras were conceived by Fomin and Zelevinsky (1) in the spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory of quivers and finite dimensional algebras, cf., for example, refs. 2⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–15; Poisson geometry (16⇓⇓–19); Teichmuller theory (20⇓⇓⇓–24); string theory (25⇓⇓⇓⇓⇓–31); discrete dynamical systems and integrability (6, 32⇓⇓⇓⇓⇓–38); and combinatorics (39⇓⇓⇓⇓⇓⇓⇓–47). Quite remarkably, cluster algebras provide a unifying algebraic and combinatorial framework for a wide variety of phenomena in these and other settings. We refer the reader to the survey papers (36, 48⇓⇓⇓⇓–53) and to the cluster algebras portal (www.math.lsa.umich.edu/~fomin/cluster.html) for various introductions to cluster algebras and their links with other subjects in mathematics (and physics). In brief, a cluster algebra A of rank k is a subring of an ambient field ℱ of rational functions in k variables, say x 1, …, x k . Unlike most commutative rings, a cluster algebra is not presented at the outset via a complete set of generators and relations. Instead, from the data of the initial seed — which includes the k initial cluster variables x 1, …, x k , plus an exchange matrix — one uses an iterative procedure called “mutation” to produce the rest of the cluster variables. In particular, each new cluster … [↵][1]1To whom correspondence should be addressed. Email: williams{at}math.berkeley.edu. [1]: #xref-corresp-1-1

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.

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.

In this paper we study the generating function f(t) for the sequence of the moments R P i (z)q(z)dz, i > 0, where P(z), q(z) are rational functions of one complex variable and is a curve in C. We calculate an analytical expression for f(t) and provide conditions imply- ing that f(t) is rational or vanishes identically. In particular, for P(z) in generic position we give an explicit criterion for a function q(z) to be orthogonal to all powers of P(z) on . As an application, we prove a stronger form of the Wermer theorem, describing analytic functions sat- isfying the system of equations R S1 h i (z)g j (z)g ' (z)dz = 0, i > 0, j > 0, in the case where the functions h(z), g(z) are rational. We also gen- eralize the theorem of Duistermaat and van der Kallen about Laurent polynomials L(z) whose integer positive powers have no constant term, and prove other results about Laurent polynomials L(z), m(z) satisfying R S1 L

Jim Propp recently introduced a variant of chip-firing on a line where the chips are given distinct integer labels. Hopkins, McConville, and Propp showed that this process is confluent from some (but not all) initial configurations of chips. We recast their set-up in terms of root systems: labeled chip-firing can be seen as a root-firing process which allows the moves $\lambda \to \lambda + \alpha$ for $\alpha\in \Phi^{+}$ whenever $\langle\lambda,\alpha^\vee\rangle = 0$, where $\Phi^{+}$ is the set of positive roots of a root system of Type A and $\lambda$ is a weight of this root system. We are thus motivated to study the exact same root-firing process for an arbitrary root system. Actually, this central root-firing process is the subject of a sequel to this paper. In the present paper, we instead study the interval root-firing processes determined by $\lambda \to \lambda + \alpha$ for $\alpha\in \Phi^{+}$ whenever $\langle\lambda,\alpha^\vee\rangle \in [-k-1,k-1]$ or $\langle\lambda,\alpha^\vee\rangle \in [-k,k-1]$, for any $k \geq 0$. We prove that these interval-firing processes are always confluent, from any initial weight. We also show that there is a natural way to consistently label the stable points of these interval-firing processes across all values of $k$ so that the number of weights with given stabilization is a polynomial in $k$. We conjecture that these Ehrhart-like polynomials have nonnegative integer coefficients.

In the paper by F Calogero and the author [Commun. Math. Phys. 59 (1978), 109– 116] the formula for frequencies of small oscillations of the Sutherland system (A l case) was found. In the present note the generalization of this formula for the case of arbitrary root system is given.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Parabolic subgroup orbits on finite root systems

We give very precise bounds for the congruence subgroup growth of arithmetic groups. This allows us to determine the subgroup growth of irreducible lattices of semisimple Lie groups. In the most general case our results depend on the Generalized Riemann Hypothesis for number fields but we can state the following unconditional theorem: Let $G$ be a simple Lie group of real rank at least 2, different than $D_4(\bbc)$, and let $\Gamma$ be any non-uniform lattice of $G$. Let $s_n(\Gamma)$ denote the number of subgroups of index at most $n$ in $\Gamma$. Then the limit $\lim\limits_{n\to \infty} \frac{\log s_n(\Gamma)}{(\log n)^2/ \log \log n}$ exists and equals a constant $\gamma(G)$ which depends only on the Lie type of $G$ and can be easily computed from its root system.

We produce a short and elementary algorithm to compute an upper bound for the canonical dimension of a spit semisimple linear algebraic group. Using this algorithm we confirm previously known bounds by Karpenko [6–9] and Devyatov [5] as well as we produce new bounds (e.g., for groups of types $F_4$, adjoint $E_6$, for some semisimple groups).

Whereas discrete groups mainly describe the symmetries of regular geometric structures (crystals), continuous groups are essential in discussing the properties of particles, fields (atoms and all the more elementary particles) and conservation laws. We restrict the investigation here to Lie groups and the Lie algebras connected with them. First we discuss the fundamental notions and relations for these groups, which are the generators, the (unitary) REPs, the invariants, connected spaces, covering and compact groups, simple and semisimple groups, etc. Most of the ideas are illustrated with the ℒU(n) groups. The central notions like symmetry projection operators, Clebsch-Gordan decomposition and the Wigner-Eckart theorem known from finite groups are generalized to continuous ones. For the investigation of semisimple groups and their IRs the knowledge of their weight and root systems is extremely useful.KeywordsStructure ConstantMaximal WeightCasimir OperatorWeight SystemInfinitesimal GeneratorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A Weyl group multiple Dirichlet seriesis a Dirichlet series in several complex variables attached to a root system Φ. The number of variables equals the rank rof the root system, and the series satisfies a group of functional equations isomorphic to the Weyl group Wof Φ. In this paper we construct a Weyl group multiple Dirichlet series over the rational function field using n th order Gauss sums attached to the root system of type A 2. The basic technique is that of [11, 10]; namely, we construct a rational function in rvariables invariant under a certain action of W, and use this to build a “local factor” of the global series.

The theory of water uptake by a single root is reviewed, including infinite and finite systems, and constant and variable diffusivity. This theory predicts the water potential and content at the root surface will depend upon soil and root parameters, most of which are readily determinable, but the uptake rate of water per unit length of root is uncertain. Uptake rates averaged over whole root systems will usually be small, suggesting only very small gradients of water content around roots, but it is likely that local rates may, on occasion, be much larger than the averages, and the possibility of large gradients cannot then be dismissed. Certain special effects may alter these conclusions, particularly salt accumulation around roots and no contact or partial contact of roots with soil. Both demand a fast uptake rate of water before they become important, and salt accumulation is unlikely to be important except in saline soils. Methods of dealing with whole root systems are briefly considered. Differing root density can be dealt with by subdividing the root volume into compartments, or by regarding the problem as one of dispersion pattern. Further advance requires better information about local boundary conditions imposed by roots on the surrounding soil.

Cluster algebras are commutative algebras that were introduced by Fomin and Zelevinsky in order to model the dual canonical basis of a quantum group and total positivity in algebraic groups. Cluster categories were introduced as a representation-theoretic model for cluster algebras. In this article we use this representation-theoretic approach to prove a conjecture of Fomin and Zelevinsky, that for cluster algebras with no coefficients associated to quivers with no oriented cycles, a seed is determined by its cluster. We also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.