This content will become publicly available on July 1, 2025
 NSFPAR ID:
 10513377
 Publisher / Repository:
 Springer Nature
 Date Published:
 Journal Name:
 Selecta Mathematica
 Volume:
 30
 Issue:
 3
 ISSN:
 10221824
 Page Range / eLocation ID:
 56
 Subject(s) / Keyword(s):
 Grothendieck polynomials Determinantal processes Random partitions Limit shape Biorthogonal ensembles Principal minor assignment problem
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

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 compute the Euler characteristic of the structure sheaf of the Brill–Noether locus of linear series with special vanishing at up to two marked points. When the Brill–Noether number $\rho $ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $\rho =1$, we recover the formulas of EisenbudHarris, Pirola, and Chan–Martín–Pflueger–Teixidor for the arithmetic genus of a Brill–Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the $K$theory class of certain degeneracy loci. Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321avoiding permutations and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey–Jockusch–Stanley expressing Schubert polynomials for 321avoiding permutations as generating functions for flagged skew tableaux.more » « less

Abstract We introduce and study a one parameter deformation of the polynuclear growth (PNG) in (1+1)dimensions, which we call the $t$PNG model. It is defined by requiring that, when two expanding islands merge, with probability $t$ they sprout another island on top of the merging location. At $t=0$, this becomes the standard (nondeformed) PNG model that, in the droplet geometry, can be reformulated through longest increasing subsequences of uniformly random permutations or through an algorithm known as patience sorting. In terms of the latter, the $t$PNG model allows errors to occur in the sorting algorithm with probability $t$. We prove that the $t$PNG model exhibits onepoint Tracy–Widom Gaussian Unitary Ensemble asymptotics at large times for any fixed $t\in [0,1)$, and onepoint convergence to the narrow wedge solution of the Kardar–Parisi–Zhang equation as $t$ tends to $1$. We further construct distributions for an external source that are likely to induce Baik–Ben Arous–Péchétype phase transitions. The proofs are based on solvable stochastic vertex models and their connection to the determinantal point processes arising from Schur measures on partitions.more » « less

Abstract We study the asymptotic limit of random pure dimer coverings on rail yard graphs when the mesh sizes of the graphs go to 0. Each pure dimer covering corresponds to a sequence of interlacing partitions starting with an empty partition and ending in an empty partition. Under the assumption that the probability of each dimer covering is proportional to the product of weights of present edges, we obtain the limit shape (law of large numbers) of the rescaled height functions and the convergence of the unrescaled height fluctuations to a diffeomorphic image of the Gaussian free field (Central Limit Theorem), answering a question in [7]. Applications include the limit shape and height fluctuations for pure steep tilings [9] and pyramid partitions [20; 36; 39; 38]. The technique to obtain these results is to analyze a class of Macdonald processes which involve dual partitions as well.

null (Ed.)Abstract We investigate the longstanding problem of finding a combinatorial rule for the Schubert structure constants in the $K$theory of flag varieties (in type $A$). The Grothendieck polynomials of A. Lascoux–M.P. Schützenberger (1982) serve as polynomial representatives for $K$theoretic Schubert classes; however no positive rule for their multiplication is known in general. We contribute a new basis for polynomials (in $n$ variables) which we call glide polynomials, and give a positive combinatorial formula for the expansion of a Grothendieck polynomial in this basis. We then provide a positive combinatorial Littlewood–Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. Our techniques easily extend to the $\beta$Grothendieck polynomials of S. Fomin–A. Kirillov (1994), representing classes in connective $K$theory, and we state our results in this more general context.more » « less