arXiv subjects
Publications and source records attributed to Paul Zinn-Justin.
This is a review (including some background material) of the author's work and related activity on certain exactly solvable statistical models in two dimensions, including the six-vertex model, loop models and lozenge tilings. Applications to enumerative combinatorics and to algebraic geometry are described.
We study the nested permutation classes $Av(4123,4231,4312)\subset Av(4123,4312)$. We relate the corresponding generating functions $D^{(2)}(z)$, $D^{(3)}(z)$ to $q$-difference equations. Solving these equations allows to show that the generation functions are not D-finite, and that $[z^n]D^{(3)}(z)\sim 0.0189672325071988\ldots\, (4.46840899160393\ldots)^n$, $[z^n]D^{(2)}(z)\sim0.0849833632890843\ldots\, (3+2\sqrt2)^n n^{-3/2}$.
We present a new method for constructing Hadamard matrices that combines transformer neural networks with local search in the PatternBoost framework. Our approach is designed for extremely sparse combinatorial search problems and is particularly effective for Hadamard matrices of Goethals--Seidel type, where Fourier methods permit fast scoring and optimisation. For orders between 100 and 200, it produces large numbers of inequivalent Hadamard matrices, and for larger orders, it succeeds where local search from random initialisation fails. The largest example found by our method has order 252. In addition to these new constructions, our experiments reveal that the transformer can discover and exploit useful hidden symmetry in the search space.
We study higher spin (pure and mixed spin) representations of the Yangian of $\mathfrak{sl}_2$. We provide a geometric realization in terms of the critical cohomology of representations of the quiver with potential of Bykov and Zinn-Justin [BZJ20]. When the framing dimension is 1, it recovers the evaluation pullback of the $\ell+1$-dimensional irreducible representation of $\mathfrak{sl}_2$. We introduce the lattice model and prove that its partition function coincides with the weight function constructed using the framed shuffle formula. The latter follows the approach of Rimányi, Tarasov and Varchenko [RTV15].
We introduce a new family of symmetric polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_λ$ arising from exactly solvable lattice models associated with the quantised loop algebra $\mathcal{U}_{q}(\mathfrak{sl}_{2}[z^\pm])$. The polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_λ$ unify $q$-Whittaker polynomials, inhomogeneous $q$-Whittaker polynomials, Grothendieck polynomials and their duals. Using Yang--Baxter equation, we derive Cauchy identities and combinatorial formulas for the transition coefficients.
In [KU23] were introduced hybrid pipe dreams interpolating between classic and bumpless pipe dreams, each hybridization giving a different formula for double Schubert polynomials. A bijective proof was given (following [GH23]) of the independence of hybridization, but only for nonequivariant Schubert polynomials. In this paper we further generalize to hybrid generic pipe dreams, replacing the bijective proof of hybridization-independence with a Yang-Baxter-based proof that allows one to maintain equivariance. An additional YB-based proof establishes a divided-difference type recurrence for these generic pipe dream polynomials. These polynomials compute something richer than double Schubert polynomials, namely the equivariant classes of the lower-upper varieties introduced in [Knu05]. We give two proofs of this: the easier being a proof that the recurrence relation holds on those classes, the more difficult being a degeneration of the lower-upper variety to a union of quadratic complete intersections (plus, possibly, some embedded components) whose individual classes match those of the generic pipe dreams. One new feature of the generic situation is a definition of the "flux" through an edge of the matrix; the notion of pipe dream itself can then be derived from the equalities among the fluxes.
The puzzle rules for computing Schubert calculus on $d$-step flag manifolds, proven in [Knutson Tao 2003] for $1$-step, in [Buch Kresch Purbhoo Tamvakis 2016] for $2$-step, and conjectured in [Coskun Vakil 2009] for $3$-step, lead to vector configurations (one vector for each puzzle edge label) that we recognize as the weights of some minuscule representations. The $R$-matrices of those representations (which, for $2$-step flag manifolds, involve triality of $D_4$) degenerate to give us puzzle formulae for two previously unsolved Schubert calculus problems: $K_T(2$-step flag manifolds$)$ and $K(3$-step flag manifolds$)$. The $K(3$-step flag manifolds$)$ formula, which involves 151 new puzzle pieces, implies Buch's correction to the first author's 1999 conjecture for $H^*(3$-step flag manifolds$)$.
We provide a combinatorial formula for the structure constants of spin Hall--Littlewood functions. This is achieved by representing these functions and the structure constants as the partition function of a lattice model and applying the underlying Yang--Baxter equation. Our combinatorial expression is in terms of generalised honeycombs; the latter were introduced by Knutson and Tao for ordinary Littlewood--Richardson coefficients and applied to the computation of Hall polynomials by Zinn--Justin.
The exceptional series is a finite list of points on a projective line with a simple Lie algebra attached to each point. This list of Lie algebras includes the five exceptional Lie algebras. We give a uniform trigonometric $R$-matrix for the exceptional series in the representation $L\oplus I$, where $L$ is the quantum deformation of the adjoint representation and $I$ is the trivial representation. We construct a sixteen dimensional algebra, $A^\square(\mathit2)$, which interpolates the algebras $\mathrm{End}(\otimes^2(L\oplus I))$ and a 287 dimensional algebra, $A^\square(\mathit3)$, which interpolates the algebras $\mathrm{End}(\otimes^3(L\oplus I))$. The $R$-matrix lives in $A^\square(\mathit2)$ and satisfies the Yang-Baxter equation in $A^\square(\mathit3)$; it interpolates the trigonometric $R$-matrices for the points in the exceptional series.
We recall the lower-upper varieties from [Knutson '05] and give a formula for their equivariant cohomology classes, as a sum over generic pipe dreams. We recover as limits the classic and bumpless pipe dream formulae for double Schubert polynomials. As a byproduct, we obtain a formula for the degree of the $n$th commuting variety as a sum of powers of 2. Generic pipe dreams also appear in the Segre-Schwarz-MacPherson analogue of the AJS/Billey formula, and when computing the Chern-Schwarz-MacPherson class of the orbit $B_- w B_+ \subseteq Mat_{k\times n}$ or of a double Bruhat cell $B_-u B_+ \cap B_+ v B_-$.
We discuss the use of methods coming from integrable systems to study problems of enumerative and algebraic combinatorics, and develop two examples: the enumeration of Alternating Sign Matrices and related combinatorial objects, and the theory of symmetric polynomials.
In Schubert Puzzles and Integrability I we proved several "puzzle rules" for computing products of Schubert classes in K-theory (and sometimes equivariant K-theory) of d-step flag varieties. The principal tool was "quantum integrability", in several variants of the Yang--Baxter equation; this let us recognize the Schubert structure constants as q->0 limits of certain matrix entries in products of R- (and other) matrices of quantized affine algebra representations. In the present work we give direct cohomological interpretations of those same matrix entries but at finite q: they compute products of "motivic Segre classes", closely related to K-theoretic Maulik--Okounkov stable classes living on the cotangent bundles of the flag varieties. Without q->0, we avoid some divergences that blocked fuller understanding of d=3,4. The puzzle computations are then explained (in cohomology onlyin this work, not K-theory) in terms of Lagrangian convolutions between Nakajima quiver varieties. More specifically, the conormal bundle to the diagonal inclusion of a flag variety factors through a quiver variety that is not a cotangent bundle, and it is on that intermediate quiver variety that the R-matrix calculation occurs.
In paper I of this series we gave positive formulae for expanding the product $\mathfrak S^π\mathfrak S^ρ$ of two Schubert polynomials, in the case that both $π,ρ$ had shared descent set of size $\leq 3$. Here we introduce and give positive formulae for two new classes of Schubert product problems: separated descent in which $π$'s last descent occurs at (or before) $ρ$'s first, and almost separated descent in which $π$'s last two descents occur at (or before) $ρ$'s first two respectively. In both cases our puzzle formulae extend to $K$-theory (multiplying Grothendieck polynomials), and in the separated descent case, to equivariant $K$-theory. The two formulae arise (via quantum integrability) from fusion of minuscule quantized loop algebra representations in types $A$, $D$ respectively.
The commutative trigonometric shuffle algebra ${\mathrm A}$ is a space of symmetric rational functions satisfying certain wheel conditions. We describe a ring isomorphism between ${\mathrm A}$ and the center of the Hecke algebra using a realization of the elements of ${\mathrm A}$ as partition functions of coloured lattice paths associated to the $R$-matrix of $\mathcal U_{t^{1/2}}(\widehat{gl}_{\infty})$. As an application, we compute under certain conditions the Hilbert series of the commuting scheme and identify it with a particular element of the shuffle algebra ${\mathrm A}$, thus providing a combinatorial formula for it as a "domain wall" type partition function of coloured lattice paths.
We study solvable lattice models associated to canonical Grothendieck polynomials and their duals. We derive inversion relations and Cauchy identities.
We address the six vertex model on a random lattice, which in combinatorial terms corresponds to the enumeration of weighted 4-valent planar maps equipped with an Eulerian orientation. This problem was exactly, albeit non-rigorously solved by Ivan Kostov in 2000 using matrix integral techniques. We convert Kostov's work to a combinatorial argument involving functional equations coming from recursive decompositions of the maps, which we solve rigorously using complex analysis. We then investigate modular properties of the solution, which lead to simplifications in certain special cases. In particular, in two special cases of combinatorial interest we rederive the formulae discovered by Bousquet-Mélou and the first author.
An expression for the R-matrix associated to $U_q(\widehat{e_8})$ in its 249-dimensional representation is given using the diagrammatic calculus of $U_q(e_8)$ invariants.
We compute the rational $\mathfrak{sl}_2$ $R$-matrix acting in the product of two spin-$\ell\over 2$ (${\ell \in \mathbb{N}}$) representations, using a method analogous to the one of Maulik and Okounkov, i.e., by studying the equivariant (co)homology of certain algebraic varieties. These varieties, first considered by Nekrasov and Shatashvili, are typically singular. They may be thought of as the higher spin generalizations of $A_1$ Nakajima quiver varieties (i.e., cotangent bundles of Grassmannians), the latter corresponding to $\ell=1$.