The tensor powers of the vector representation associated to an infinite rank quantum group decompose into irreducible components with multiplicities independant of the infinite root system considered. Although the irreducible modules obtained in this way are not of highest weight, they admit a crystal basis and a canonical basis. This permits in particular to obtain for each familly of classical Lie algebras a Robinson-Schensted correspondence on biwords defined on infinite alphabets. We then depict a structure of bi-crystal on these bi-words. This RS-correspondence yields also a plactic algebra and plactic Schur functions distinct for each infinite root system. Surprisingly, the algebras spanned by these plactic Schur functions are all isomorphic to the algebra of symmetric functions.
Motivated by a question on the graded rank of the stalks of the canonical sheaf on a Bruhat graph, we lift some equalities concerning (parabolic) Kazhdan-Lusztig polynomials to this moment graph setting. Our proofs hold also in positive characteristic, under some technical assumptions.
For a $\{0,1\}$-valued matrix $M$ let $\rm{CC}(M)$ denote the deterministic communication complexity of the boolean function associated with $M$. The log-rank conjecture of Lovász and Saks [FOCS 1988] states that $\rm{CC}(M) \leq \log^c(\rm{rank}(M))$ for some absolute constant $c$ where $\rm{rank}(M)$ denotes the rank of $M$ over the field of real numbers. We show that $\rm{CC}(M)\leq c \cdot \rm{rank}(M)/\log \rm{rank}(M)$ for some absolute constant $c$, assuming a well-known conjecture from additive combinatorics known as the Polynomial Freiman-Ruzsa (PFR) conjecture. Our proof is based on …
The category of Cohen-Macaulay modules of an algebra $B_{k,n}$ is used [JKS16] to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. In this paper, we find canonical Auslander--Reiten sequences and study the Auslander--Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen-Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the assoc…
Let A be an integer (d x n) matrix, and assume that the convex hull conv(A) of its columns is a simplex of dimension d-1. Write \NA for the semigroup generated by the columns of A. It was proved by M. Saito [math.AG/0012257] that the semigroup ring \CC[\NA] over the complex numbers \CC is Cohen-Macaulay if and only if the rank of the GKZ hypergeometric system H_A(beta) equals the normalized volume of conv(A) for all complex parameters beta in \CC^d. Our refinement here shows, in this simplicial case, that H_A(beta) has rank strictly larger than the volume of conv(A) if and only if beta lies in…
Mirkovic and Vilonen discovered a canonical basis of algebraic cycles for the intersection homology of (the closures of the strata of) the loop Grassmannian. The moment map images of these varieties are a collection of polytopes, and they may be used to compute weight multiplicities and tensor product multiplicities for representations of a semisimple group. The polytopes are explicitly described for a few low rank groups.
Occam's Razor tells us to pick the simplest model that fits our observations. In order to make sense of his process mathematically, we interpret it in the context of posets of functions. Our approach leads to some unusual new combinatorial problems concerning functions between finite sets. The same ideas are used to define a nicely behaved and apparently unknown analogue of the rank of a group. We also make a construction that associates with each group an infinite sequence of numbers called its fusion sequence. The first term in this sequence is determined by the rank of the group and we prov…
The cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. For certain monoids connected with combinatorics, such as the plactic monoid (the monoid of Young tableaux) and the sylvester monoid (the monoid of binary search trees), connected components consist of elements that have the same evaluation (that is, contain the same number of each generating symbol). This paper discusses new results on the diameters of connected components of the cyclic shift graphs of the finite-rank analogues of these monoids…
Let $G$ be a non-compact classical semisimple Lie group and let $G/V$ be the adjoint orbit with respect to a fixed element in $G$. These manifolds can be equipped with an almost-Kähler structure and we provide explicit formulae for the existence of special almost-complex structures on $G/V$ purely in terms of the combinatorics of the associated Vogan diagram. The formulae are given separately for Lie groups whose Lie algebras are of type $A_{\ell}$, $B_{\ell}$, $C_{\ell}$, $D_{\ell}$, where $\ell$ denotes the rank of the Lie algebra.
We introduce a general class of regular weight functions on finite abelian groups, and study the combinatorics, the duality theory, and the metric properties of codes endowed with such functions. The weights are obtained by composing a suitable support map with the rank function of a graded lattice satisfying certain regularity properties. A regular weight on a group canonically induces a regular weight on the character group, and invertible MacWilliams identities always hold for such a pair of weights. Moreover, the Krawtchouk coefficients of the corresponding MacWilliams transformation have …
Let U be the quantum group associated to a Lie algebra g of rank n. The negative part U^- of U has a canonical basis B with favourable properties, introduced by Kashiwara and Lusztig. The approaches of Kashiwara and Lusztig lead to a set of alternative parametrizations of the canonical basis, one for each reduced expression for the longest word in the Weyl group of g. We describe the authors' recent work establishing close relationships between the Lusztig cones, canonical basis elements and the regions of linearity of reparametrization functions arising from the above parametrizations in type…
We give a formula for the crystal structure on the integer points of the string polytopes and the $*$-crystal structure on the integer points of the string cones of type $A$ for arbitrary reduced words. As a byproduct we obtain defining inequalities for Nakashima-Zelevinsky string polytopes. Furthermore, we give an explicit description of the Kashiwara $*$-involution on string data for a special choice of reduced word.
Several bases of the Garsia-Haiman modules for hook shapes are given, as well as combinatorial decomposition rules for these modules. These bases and rules extend the classical ones for the coinvariant algebra of type $A$. We also give a decomposition of the Garsia-Haiman modules into descent representations.
For a rank two root system and a pair of nonnegative integers, using only elementary combinatorics we construct two posets. The constructions are uniform across the root systems A1+A1, A2, C2, and G2. Examples appear in Figures 3.2 and 3.3. We then form the distributive lattices of order ideals of these posets. Corollary 5.4 gives elegant quotient-of-products expressions for the rank generating functions of these lattices (thereby providing answers to a 1979 question of Stanley). Also, Theorem 5.3 describes how these lattices provide a new combinatorial setting for the Weyl characters of repre…
The cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. This paper examines the cyclic shift graphs of `plactic-like' monoids, whose elements can be viewed as combinatorial objects of some type: aside from the plactic monoid itself (the monoid of Young tableaux), examples include the hypoplactic monoid (quasi-ribbon tableaux), the sylvester monoid (binary search trees), the stalactic monoid (stalactic tableaux), the taiga monoid (binary search trees with multiplicities), and the Baxter monoid (pairs …
We give a combinatorial evaluation of Iwahori Whittaker functions for unramified genuine principal series representations on metaplectic covers of the general linear group over a non-archimedean local field. To describe the combinatorics, we introduce new combinatorial data that we call colored data: colored Lusztig data, colored Gelfand-Tsetlin patterns, and colored lattice models. We show that all three are equivalent. To achieve the result, we give an explicit Iwahori decomposition for the maximal unipotent subgroup of a split reductive group which gives the parametrization of the generaliz…
We recast the classical notion of standard tableaux in an alcove-geometric setting and extend these classical ideas to all reduced paths in our geometry. This broader path-perspective is essential for implementing the higher categorical ideas of Elias--Williamson in the setting of quiver Hecke algebras. Our first main result is the construction of light leaves bases of quiver Hecke algebras. These bases are richer and encode more structural information than their classical counterparts, even in the case of the symmetric groups. Our second main result provides path-theoretic generators for the …
In this paper, we continue the development of a new combinatorial model for the irreducible characters of a complex semisimple Lie group. This model, which will be referred to as the alcove path model, can be viewed as a discrete counterpart to the Littelmann path model. It leads to an extensive generalization of the combinatorics of irreducible characters from Lie type A (where the combinatorics is based on Young tableaux, for instance) to arbitrary type. The main results of this paper are: (1) a combinatorial description of the crystal graphs corresponding to the irreducible representations …
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进学会数学研究所）；Institute for Advanced Study, Massey University（梅西大学高等研究院）；Department of Mathematics, The University of Auckland（奥克兰大学数学系）
We explore combinatorics associated with the degenerate Hecke algebra at $q=0$, obtaining a formula for a system of orthogonal idempotents, and also exploring various pattern avoidance results. Generalizing constructions for the 0-Hecke algebra, we explore the representation theory of $\JJ$-trivial monoids. We then discuss two-tensors of crystal bases for $U_q(\tilde{\mathfrak{sl}_2})$, establishing a complementary result to one of Bandlow, Schilling, and Thiéry on affine crystals arising from promotion operators. Finally, we give a computer implementation of Stembridge's local axioms for simp…
We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type $A$. Using geometric objects called Rhombic tilings we derive a "crossing formula" to compute the actions of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element. We provide the following three applications of this result. Using the tropical Chamber Ansatz of Berenstein-Fomin-Zelevinsky we prove an enhanced version of the Anderson-Mirković conjecture for the crystal structure on MV polytopes.…
We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves less products of building blocks of multinomial type, and we study the combinatorics of the coefficients showing up in both formulae.
This paper uses the combinatorics of Young tableaux to prove the plactic monoid of infinite rank does not satisfy a non-trivial identity, by showing that the plactic monoid of rank $n$ cannot satisfy a non-trivial identity of length less than or equal to $n$. A new identity is then proven to hold for the monoid of $n \times n$ upper-triangular tropical matrices. Finally, a straightforward embedding is exhibited of the plactic monoid of rank $3$ into the direct product of two copies of the monoid of $3\times 3$ upper-triangular tropical matrices, giving a new proof that the plactic monoid of ra…
We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type $\tilde{C}$, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for both algebras. The elements of these bases are parameterized by certain subsets of fully commutative elements. We enumerate these elements according to their affine length.
Chuang and Rouquier describe an action by perverse equivalences on the set of bases of a triangulated category of Calabi-Yau dimension $-1$. We develop an analogue of their theory for Calabi-Yau categories of dimension $w<0$ and show it is equivalent to the mutation theory of $w$-simple-minded systems. Given a non-positively graded, finite-dimensional symmetric algebra $A$, we show that the differential graded stable category of $A$ has negative Calabi-Yau dimension. When $A$ is a Brauer tree algebra, we construct a combinatorial model of the dg-stable category and show that perverse equivalen…
We generalise surface cluster algebras to the case of infinite surfaces where the surface contains finitely many accumulation points of boundary marked points. To connect different triangulations of an infinite surface, we consider infinite mutation sequences. We show transitivity of infinite mutation sequences on triangulations of an infinite surface and examine different types of mutation sequences. Moreover, we use a hyperbolic structure on an infinite surface to extend the notion of surface cluster algebras to infinite rank by giving cluster variables as lambda lengths of arcs. Furthermore…
We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras, higher homotopy groups, cohomology rings, twisted homology with rank 1 complex coefficients, and Milnor fibers.
A number of finite algorithms for constructing representation theoretic data from group multiplications in a finite group G have recently been shown to be related to amplitudes for combinatoric topological strings (G-CTST) based on Dijkgraaf-Witten theory of flat G-bundles on surfaces. We extend this result to projective representations of G using twisted Dijkgraaf-Witten theory. New algorithms for characters are described, based on handle creation operators and minimal multiplicative generating subspaces for the centers of group algebras and twisted group algebras. Such minimal generating sub…
Group algebras of permutations have proved highly useful in solving a number of problems in large N gauge theories. I review the use of permutations in classifying gauge invariants in one-matrix and multi-matrix models and computing their correlators. These methods are also applicable to tensor models and have revealed a link between tensor models and the counting of branched covers. The key idea is to parametrize $U(N)$ gauge invariants using permutations, subject to equivalences. Correlators are related to group theoretic properties of these equivalence classes. Fourier transformation on sym…
We study $r$-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including the Young lattice. We first provide a simple bijection relating differential posets to a certain class of hypergraphs, including all finite projective planes, which are shown to be naturally embedded in the initial ranks of some differential poset. As a byproduct, we prove the existence, if and only if $r\geq 6$, of $r$-differential posets no…
We give lower bounds for the rank of the first homology group of the real points of the Deligne-Mumford-Knudsen compactification of stable n-pointed curves of genus 0,which coincides with the Chow quotient (RP^1)^n//PGL(2,R).The study has connections with spectral geometry and combinatorics.
In this paper we introduce a binomial ideal derived from a binary linear code. We present some applications of a Gröbner basis of this ideal with respect to a total degree ordering. In the first application we give a decoding method for the code. By associating the code with the set of cycles in a graph, we can solve the problem of finding all codewords of minimal length (minimal cycles in a graph), and show how to find a minimal cycle basis. Finally we discuss some results on the computation of the Gröbner basis.
We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstra…
It is a classical result of Stein and Waterman that the asymptotic number of RNA secondary structures is $1.104366 \cdot n^{-3/2} \cdot 2.618034^n$. Motivated by the kinetics of RNA secondary structure formation, we are interested in determining the asymptotic number of secondary structures that are locally optimal, with respect to a particular energy model. In the Nussinov energy model, where each base pair contributes -1 towards the energy of the structure, locally optimal structures are exactly the saturated structures, for which we have previously shown that asymptotically, there are $1.07…
We describe various combinatorial aspects of the Birkhoff-Connes-Kreimer factorization in perturbative renormalisation. The analog of Bogoliubov's preparation map on the Lie algebra of Feynman graphs is identified with the pre-Lie Magnus expansion. Our results apply to any connected filtered Hopf algebra, based on the pro-nilpotency of the Lie algebra of infinitesimal characters.
Hecke-Kiselman monoids $\textrm{HK}_Θ$ and their algebras $K[\textrm{HK}_Θ]$, over a field $K$, associated to finite oriented graphs $Θ$ are studied. In the case $Θ$ is a cycle of length $n\geqslant 3$, a hierarchy of certain unexpected structures of matrix type is discovered within the monoid $C_n=\textrm{HK}_Θ$ and it is used to describe the structure and the properties of the algebra $K[C_n]$. In particular, it is shown that $K[C_n]$ is a right and left Noetherian algebra, while it has been known that it is a PI-algebra of Gelfand-Kirillov dimension one. This is used to characterize all Noe…
For every finite Coxeter group $Γ$, each positive braids in the corresponding braid group admits a unique decomposition as a finite sequence of elements of $Γ$, the so-called Garside-normal form.The study of the associated adjacency matrix $Adj(Γ)$ allows to count the number of Garside-normal form of a given length.In this paper we prove that the characteristic polynomial of $Adj(B_n)$ divides the one of $Adj(B_{n+1})$. The key point is the use of a Hopf algebra based on signed permutations. A similar result was already known for the type $A$. We observe that this does not hold for type $D$. T…
The key-agreement problem (finding a private key to use for secret messages, otherwise referred to as the public-key distribution problem), was introduced by Diffie and Hellman in 1976. An approach to structuring key-agreement protocols via the use of one-way associative functions was proposed in 1993 by Rabi and Sherman. We propose here a provably strong associative one-way function based upon knot composition (answering an open problem proposed by Rabi and Sherman whether any such associative one-way functions exist). We also introduce and solve a game, exploring its relation to problems in …
We discuss three convolution inequalities that are connected to additive combinatorics. Cloninger and the second author showed that for nonnegative $f \in L^1(-1/4, 1/4)$, $$ \max_{-1/2 \leq t \leq 1/2} \int_{\mathbb{R}}{f(t-x) f(x) dx} \geq 1.28 \left( \int_{-1/4}^{1/4}{f(x) dx}\right)^2$$ which is related to $g-$Sidon sets (1.28 cannot be replaced by 1.52). We prove a dual statement, related to difference bases, and show that for $f \in L^1(\mathbb{R})$, $$ \min_{0 \leq t \leq 1}\int_{\mathbb{R}}{f(x) f(x+t) dx} \leq 0.42 \|f\|_{L^1}^2,$$ where the constant 1/2 is trivial, 0.42 cannot be rep…
We introduce bijections between generalized type $A_n$ noncrossing partitions (that is, associated to arbitrary standard Coxeter elements) and fully commutative elements of the same type. The latter index the diagram basis of the classical Temperley-Lieb algebra, while for each choice of standard Coxeter element the corresponding noncrossing partitions also index a basis, given by the images in the Temperley-Lieb algebra of the simple elements of the dual Garside structure (associated to this choice of standard Coxeter element) of the Artin braid group on $n+1$ strands. We then show that our b…
Let C be a clutter and let I be its edge ideal. We present a combinatorial description of the minimal generators of the symbolic Rees algebra Rs(I) of I. It is shown that the minimal generators of Rs(I) are in one to one correspondence with the irreducible parallelizations of C. From our description some major results on symbolic Rees algebras of perfect graphs and clutters will follow. As a byproduct, we give a method, using Hilbert bases, to compute all irreducible parallelizations of C along with all the corresponding vertex covering numbers.
We review and present new studies on the relation between the partition functions of integrable lattice models and symmetric polynomials, and its combinatorial representation theory based on the correspondence, including our work. In particular, we examine the correspondence between the wavefunctions of the XXZ type, Felderhof type and the boson type integrable models and symmetric polynomials such as the Schur, Grothendieck and symplectic Schur functions. We also give a brief report of our work on generalizing the correspondence between the Felderhof models and factorial Schur and symplectic …
