arXiv subjects
Publications and source records attributed to Anne Schilling.
We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic and in terms of the dinv and dinv of depth statistics. We prove symmetry using an involution on plane trees. The same involution proves symmetry of the Tutte polynomials. We also provide a combinatorial proof of a remark by Garsia et al. regarding parking functions and the number of connected graphs on a fixed number of vertices.
We provide a simple proof of the Holonomy Theorem using a new Lyndon-Chiswell length function on the Karnofsky-Rhodes expansion of a semigroup. Unexpectedly, we have both a left and a right action on the Chiswell tree by elliptic maps.
We review the recent approach to Markov chains using the Karnofksy-Rhodes and McCammond expansions in semigroup theory by the authors and illustrate them by two examples.
We introduce a type $A$ crystal structure on decreasing factorizations of fully-commutative elements in the 0-Hecke monoid which we call $\star$-crystal. This crystal is a $K$-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the $\star$-crystal intertwines with the crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators.
Recent work of the first author, Negut and Rasmussen, and of Oblomkov and Rozansky in the context of Khovanov--Rozansky knot homology produces a family of polynomials in $q$ and $t$ labeled by integer sequences. These polynomials can be expressed as equivariant Euler characteristics of certain line bundles on flag Hilbert schemes. The $q,t$-Catalan numbers and their rational analogues are special cases of this construction. In this paper, we give a purely combinatorial treatment of these polynomials and show that in many cases they have nonnegative integer coefficients. For sequences of length at most 4, we prove that these coefficients enumerate subdiagrams in a certain fixed Young diagram and give an explicit symmetric chain decomposition of the set of such diagrams. This strengthens results of Lee, Li and Loehr for $(4,n)$ rational $q,t$-Catalan numbers.
We generalize the Robinson-Schensted-Knuth algorithm to the insertion of two row arrays of multisets. This generalization leads to new enumerative results that have representation theoretic interpretations as decompositions of centralizer algebras and the spaces they act on. In addition, restrictions on the multisets lead to further identities and representation theory analogues. For instance, we obtain a bijection between words of length $k$ with entries in $[n]$ and pairs of tableaux of the same shape with one being a standard Young tableau of size $n$ and the other being a standard multiset tableau of content $[k]$. We also obtain an algorithm from partition diagrams to pairs of a standard tableau and a standard multiset tableau of the same shape, which has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. This insertion algorithm matches recent representation-theoretic results of Halverson and Jacobson.
We consider Jack polynomials $J_λ$ and their shifted analogue $J^#_λ$. In 1989, Stanley conjectured that $\langle J_μJ_ν, J_λ\rangle$ is a polynomial with nonnegative coefficients in the parameter $α$. In this note, we extend this conjecture to the case of shifted Jack polynomials.
We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup $S$. Our methods use geometric finite semigroup theory via the Karnofsky-Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to $P(n)$, the set of all subsets of an $n$ element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or $\mathscr{R}$-trivial bands) by Brown and Diaconis, extensions to $\mathscr{R}$-trivial semigroups by Ayyer, Steinberg, Thiéry and the second author, and important recent work by Chung and Graham. The Karnofsky-Rhodes expansion of the right Cayley graph of $S$ in terms of generators yields again a right Cayley graph. The McCammond expansion provides normal forms for elements in the expanded $S$. Using our previous results with Silva based on work by Berstel, Perrin, Reutenauer, we construct (infinite) semaphore codes on which we can define Markov chains. These semaphore codes can be lumped using geometric semigroup theory. Using normal forms and associated Kleene expressions, they yield formulas for the stationary distribution of the finite Markov chain of the expanded $S$ and the original $S$. Analyzing the normal forms also provides an estimate on the mixing time.
We show that the stationary distribution of a finite Markov chain can be expressed as the sum of certain normal distributions. These normal distributions are associated to planar graphs consisting of a straight line with attached loops. The loops touch only at one vertex either of the straight line or of another attached loop. Our analysis is based on our previous work, which derives the stationary distribution of a finite Markov chain using semaphore codes on the Karnofsky--Rhodes and McCammond expansion of the right Cayley graph of the finite semigroup underlying the Markov chain.
We provide a characterization of the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et al.. This characterization is a combination of local queer axioms generalizing Stembridge's local axioms for crystal bases for simply-laced root systems, which were recently introduced by Assaf and Oguz, with further axioms and a new graph $G$ characterizing the relations of the type $A$ components of the queer crystal. We provide a counterexample to Assaf's and Oguz' conjecture that the local queer axioms uniquely characterize the queer supercrystal. We obtain a combinatorial description of the graph $G$ on the type $A$ components by providing explicit combinatorial rules for the odd queer operators on certain highest weight elements.
We establish a bijection between rigged configurations and highest weight elements of a tensor product of Kirillov-Reshetikhin crystals for all nonexceptional types. A key idea for the proof is to embed both objects into bigger sets for simply-laced types $A_n^{(1)}$ or $D_n^{(1)}$, whose bijections have already been established. As a consequence we settle the $X=M$ conjecture in full generality for nonexceptional types. Furthermore, the bijection extends to a classical crystal isomorphism and sends the combinatorial $R$-matrix to the identity map on rigged configurations.
Combining results of T.K. Lam and J. Stembridge, the type $C$ Stanley symmetric function $F_w^C(\mathbf{x})$, indexed by an element $w$ in the type $C$ Coxeter group, has a nonnegative integer expansion in terms of Schur functions. We provide a crystal theoretic explanation of this fact and give an explicit combinatorial description of the coefficients in the Schur expansion in terms of highest weight crystal elements.
We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization $R_{n,k}$ due to Haglund, Rhoades and Shimozono of the coinvariant algebra $R_n$. The crystal structure also enables us to demonstrate the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.
Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This parallels the relationship between Schur functions and Demazure characters for the general linear group. We establish this connection by imposing a Demazure crystal structure on key tableaux, recently introduced by the first author in connection with Demazure characters and Schubert polynomials, and linking this to the type A crystal structure on reduced word factorizations, recently introduced by Morse and the second author in connection with Stanley symmetric functions.
Consider the following partial "sorting algorithm" on permutations: take the first entry of the permutation in one-line notation and insert it into the position of its own value. Continue until the first entry is 1. This process imposes a forest structure on the set of all permutations of size $n$, where the roots are the permutations starting with 1 and the leaves are derangements. Viewing the process in the opposite direction towards the leaves, one picks a fixed point and moves it to the beginning. Despite its simplicity, this "fixed point forest" exhibits a rich structure. In this paper, we consider the fixed point forest in the limit $n\to \infty$ and show using Stein's method that at a random permutation the local structure weakly converges to a tree defined in terms of independent Poisson point processes. We also show that the distribution of the length of the longest path to a leaf converges to the geometric distribution with mean $e-1$, and the length of the shortest path converges to the Poisson distribution with mean 1. In addition, the higher moments are bounded and hence the expectations converge as well.
We prove that the expected number of braid moves in the commutation class of the reduced word $(s_1 s_2 \cdots s_{n-1})(s_1 s_2 \cdots s_{n-2}) \cdots (s_1 s_2)(s_1)$ for the long element in the symmetric group $\mathfrak{S}_n$ is one. This is a variant of a similar result by V. Reiner, who proved that the expected number of braid moves in a random reduced word for the long element is one. The proof is bijective and uses X. Viennot's theory of heaps and variants of the promotion operator. In addition, we provide a refinement of this result on orbits under the action of even and odd promotion operators. This gives an example of a homomesy for a nonabelian (dihedral) group that is not induced by an abelian subgroup. Our techniques extend to more general posets and to other statistics.
We establish the equality of the specialization $E_{wλ}(x;q,0)$ of the nonsymmetric Macdonald polynomial $E_{wλ}(x;q,t)$ at $t=0$ with the graded character $\mathop{\rm gch} U_{w}^{+}(λ)$ of a certain Demazure-type submodule $U_{w}^{+}(λ)$ of a tensor product of "single-column" Kirillov--Reshetikhin modules for an untwisted affine Lie algebra, where $λ$ is a dominant integral weight and $w$ is a (finite) Weyl group element, this generalizes our previous result, that is, the equality between the specialization $P_λ(x;q,0)$ of the symmetric Macdonald polynomial $P_λ(x;q,t)$ at $t=0$ and the graded character of a tensor product of single-column Kirillov--Reshetikhin modules. We also give two combinatorial formulas for the mentioned specialization of a nonsymmetric Macdonald polynomial: one in terms of quantum Lakshmibai-Seshadri paths and the other in terms of the quantum alcove model.
We establish the equality of the specialization $P_λ(x;q,0)$ of the Macdonald polynomial at $t=0$ with the graded character $X_λ(x;q)$ of a tensor product of "single-column" Kirillov-Reshetikhin (KR) modules for untwisted affine Lie algebras. This is achieved by constructing two uniform combinatorial models for the crystals associated with the mentioned tensor products: the quantum alcove model (which is naturally associated to Macdonald polynomials), and the quantum Lakshmibai-Seshadri path model. We provide an explicit affine crystal isomorphism between the two models, and realize the energy function in both models. In particular, this gives the first proof of the positivity of the $t = 0$ limit of the symmetric Macdonald polynomial in the untwisted and non-simply-laced cases, when it is expressed as a linear combination of the irreducible characters for a finite-dimensional simple Lie subalgebra, as well as a representation-theoretic meaning of the coefficients in this expression in terms of degree functions.