Consider a lattice of n sites arranged around a ring, with the $n$ sites occupied by particles of weights $\{1,2,\ldots ,n\}$; the possible arrangements of particles in sites thus correspond to the $n!$ permutations in $S_n$. The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on $S_n$, in which two adjacent particles of weights $i<j$ swap places at rate $x_i  y_{n+1j}$ if the particle of weight $j$ is to the right of the particle of weight $i$. (Otherwise, nothing happens.) When $y_i=0$ for all $i$, the stationary distribution was conjecturally linked to Schubert polynomials [18], and explicit formulas for steady state probabilities were subsequently given in terms of multiline queues [4, 5]. In the case of general $y_i$, Cantini [7] showed that $n$ of the $n!$ states have probabilities proportional to double Schubert polynomials. In this paper, we introduce the class of evilavoiding permutations, which are the permutations avoiding the patterns $2413, 4132, 4213,$ and $3214$. We show that there are $\frac {(2+\sqrt {2})^{n1}+(2\sqrt {2})^{n1}}{2}$ evilavoiding permutations in $S_n$, and for each evilavoiding permutation $w$, we give an explicit formula for the steady state probability $\psi _w$ as a product of double Schubert polynomials. (Conjecturally, all other probabilities are proportional to a positive sum of at least two Schubert polynomials.) When $y_i=0$ for all $i$, we give multiline queue formulas for the $\textbf {z}$deformed steady state probabilities and use this to prove the monomial factor conjecture from [18]. Finally, we show that the Schubert polynomials arising in our formulas are flagged Schur functions, and we give a bijection in this case between multiline queues and semistandard Young tableaux.
 Award ID(s):
 1945212
 NSFPAR ID:
 10405706
 Date Published:
 Journal Name:
 International Mathematics Research Notices
 Volume:
 2022
 Issue:
 16
 ISSN:
 10737928
 Page Range / eLocation ID:
 12653 to 12698
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract 
We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the Ktheory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (Sel Math 7(1):57â81, 2001). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are not determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the 4 Ă 4 problem. We also propose a procedure for getting Nansonlike determinantal tests for matrices of any size n â„ 4, which appear new for n â„ 5. By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (Nucl Phys B 536:704â732, 1998), we identify Grothendieck random partitions as a crosssection of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a crosssection of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation.more » « less

We introduce families of twoparameter multivariate polynomials indexed by pairs of partitions $v,w$  {\it biaxial double} $(\beta,q)${\it Grothendieck polynomials}  which specialize at $q=0$ and $v=1$ to double $\beta$Grothendieck polynomials from torusequivariant connective Ktheory. Initially defined recursively via divided difference operators, our main result is that these new polynomials arise as partition functions of solvable lattice models. Moreover, the associated quantum group of the solvable model for polynomials in $n$ pairs of variables is a Drinfeld twist of the $U_q(\widehat{\mathfrak{sl}}_{n+1})$ $R$matrix. By leveraging the resulting YangBaxter equations of the lattice model, we show that these polynomials simultaneously generalize double $\beta$Grothendieck polynomials and dual double $\beta$Grothendieck polynomials for arbitrary permutations. We then use properties of the model and YangBaxter equations to reprove FominKirillov's Cauchy identity for $\beta$Grothendieck polynomials, generalize it to a new Cauchy identity for biaxial double $\beta$Grothendieck polynomials, and prove a new branching rule for double $\beta$Grothendieck polynomials.more » « less

Abstract We study the family of irreducible modules for quantum affine
whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to$\mathrm{\u0111\x9d\x94\xb0}\xe2\x81\u0105{\mathrm{\u0111\x9d\x94\copyright}}_{n+1}$ {\mathfrak{sl}_{n+1}} with${A}_{m}$ {A_{m}} . These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category$m\xe2\x89\u20acn$ {m\leq n} . This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) KirillovâReshetikhin module with its dual always contains an imaginary module in its JordanâHĂ¶lder series and give an explicit formula for its Drinfeld polynomial. Together with the results of [D. Hernandez and B. Leclerc,A cluster algebra approach to${\mathcal{\u0111\x9d\x92\x9e}}^{}$ {\mathscr{C}^{}} q characters of KirillovâReshetikhin modules,J. Eur. Math. Soc. (JEMS) 18 2016, 5, 1113â1159] this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of [E. Lapid and A. MĂnguez,Geometric conditions for irreducibility of certain representations of the general linear group over a nonarchimedean local field,Adv. Math. 339 2018, 113â190]. Finally, we use our methods to give a family of imaginary modules in type$\mathrm{\xe2\x96\u0104}$ \square which do not arise from an embedding of${D}_{4}$ {D_{4}} with${A}_{r}$ {A_{r}} in$r\xe2\x89\u20ac3$ {r\leq 3} .${D}_{4}$ {D_{4}} 
Abstract We say that two permutations $\pi $ and $\rho $ have separated descents at position $k$ if $\pi $ has no descents before position $k$ and $\rho $ has no descents after position $k$. We give a counting formula, in terms of reduced word tableaux, for computing the structure constants of products of Schubert polynomials indexed by permutations with separated descents, and recognize that these structure constants are certain EdelmanâGreene coefficients. Our approach uses generalizations of SchĂŒtzenbergerâs jeu de taquin algorithm and the EdelmanâGreene correspondence via bumpless pipe dreams.