We construct a novel family of difference-permutation operators and prove that they are diagonalized by the wreath Macdonald P-polynomials; the eigenvalues are written in terms of elementary symmetric polynomials of arbitrary degree. Our operators arise from integral formulas for the action of the horizontal Heisenberg subalgebra in the vertex representation of the corresponding quantum toroidal algebra.

Let $\mathcal {D}$ be a Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated category with a rigid object R. Let $\Lambda =\operatorname {End}_{\mathcal {D}}R$ be the endomorphism algebra of R. We introduce the notion of mutation of maximal rigid objects in the two-term subcategory $R\ast R[1]$ via exchange triangles, which is shown to be compatible with the mutation of support $\tau $-tilting $\Lambda $-modules. In the case that $\mathcal {D}$ is the cluster category arising from a punctured marked surface, it is shown that the graph of mutations of support $\tau $-tilting $\Lambda $-modules is isomorphic to the graph of flips of certain collections of tagged arcs on the surface, which is moreover proved to be connected. Consequently, the mutation graph of support $\tau $-tilting modules over a skew-gentle algebra is connected. This generalizes one main result in [49].

We prove a criterion of when the dual character $\chi _{D}(x)$ of the flagged Weyl module associated a diagram D in the grid $[n]\times [n]$ is zero-one, that is, the coefficients of monomials in $\chi _{D}(x)$ are either 0 or 1. This settles a conjecture proposed by Mészáros–St. Dizier–Tanjaya. Since Schubert polynomials and key polynomials occur as special cases of dual flagged Weyl characters, our approach provides a new and unified proof of known criteria for zero-one Schubert/key polynomials due to Fink–Mészáros–St. Dizier and Hodges–Yong, respectively.

Let W be a simply laced Weyl group of finite type and rank n. If W has type $E_7$, $E_8$ or $D_n$ for n even, then the root system of W has subsystems of type $nA_1$. This gives rise to an irreducible Macdonald representation of W spanned by n-roots, which are products of n orthogonal roots in the symmetric algebra of the reflection representation. We prove that in these cases, the set of all maximal sets of orthogonal positive roots has the structure of a quasiparabolic set in the sense of Rains–Vazirani. The quasiparabolic structure can be described in terms of certain quadruples of orthogonal positive roots which we call crossings, nestings and alignments. This leads to nonnesting and noncrossing bases for the Macdonald representation, as well as some highly structured partially ordered sets. We use the $8$-roots in type $E_8$ to give a concise description of a graph that is known to be non-isomorphic but quantum isomorphic to the orthogonality graph of the $E_8$ root system.

We give an elementary approach utilizing only the divided difference formalism for obtaining expansions of Schubert polynomials that are manifestly nonnegative, by studying solutions to the equation $\sum Y_i\partial _i=\operatorname {id}$ on polynomials with no constant term. This in particular recovers the pipe dream and slide polynomial expansions. We also show that slide polynomials satisfy an analogue of the divided difference formalisms for Schubert polynomials and forest polynomials, which gives a simple method for extracting the coefficients of slide polynomials in the slide polynomial decomposition of an arbitrary polynomial.

We introduce a family of polynomials, which arise in three distinct ways: in the large N expansion of a matrix integral, as a weighted enumeration of factorizations of permutations, and via the topological recursion. More explicitly, we interpret the complex Grassmannian $\mathrm {Gr}(M,N)$ as the space of $N \times N$ idempotent Hermitian matrices of rank M and develop a Weingarten calculus to integrate products of matrix elements over it. In the regime of large N and fixed ratio $\frac {M}{N}$, such integrals have expansions whose coefficients count factorizations of permutations into monotone sequences of transpositions, with each sequence weighted by a monomial in $t = 1 - \frac {N}{M}$. This gives rise to the desired polynomials, which specialise to the monotone Hurwitz numbers when $t = 1$. These so-called deformed monotone Hurwitz numbers satisfy a cut-and-join recursion, a one-point recursion, and the topological recursion. Furthermore, we conjecture on the basis of overwhelming empirical evidence that the deformed monotone Hurwitz numbers are real-rooted polynomials whose roots satisfy remarkable interlacing phenomena. An outcome of our work is the viewpoint that the topological recursion can be used to “topologise” sequences of polynomials, and we claim that the resulting families of polynomials may possess interesting properties.

A spline is an assignment of polynomials to the vertices of a graph whose edges are labeled by ideals, where the difference of two polynomials labeling adjacent vertices must belong to the corresponding ideal. The set of splines forms a ring. We consider spline rings where the underlying graph is the Cayley graph of a symmetric group generated by a collection of transpositions. These rings generalize the GKM construction for equivariant cohomology rings of flag, regular semisimple Hessenberg and permutohedral varieties. These cohomology rings carry two actions of the symmetric group $S_n$ whose graded characters are both of general interest in algebraic combinatorics. In this paper, we generalize the graded $S_n$-representations from the cohomologies of the above varieties to splines on Cayley graphs of $S_n$ and then (1) give explicit module and ring generators for whenever the $S_n$-generating set is minimal, (2) give a combinatorial characterization of when graded pieces of one $S_n$-representation is trivial, and (3) compute the first degree piece of both graded characters for all generating sets.

We give an explicit quadratic Gröbner basis for generalized Chow rings of supersolvable built lattices, with the help of the operadic structure on geometric lattices introduced in a previous article. This shows that the generalized Chow rings associated to minimal building sets of supersolvable lattices are Koszul. As another consequence, we get that the cohomology algebras of the components of the extended modular operad in genus $0$ are Koszul.

We give a crystal structure on the set of Gelfand–Tsetlin patterns (GTPs), which parametrize bases for finite-dimensional irreducible representations of the general linear Lie algebra. The crystal data are given in closed form and are expressed using tropical polynomial functions of the entries of the patterns. We prove that with this crystal structure, the natural bijection between GTPs and semistandard Young tableaux is a crystal isomorphism.

The Newell–Littlewood (NL) numbers are tensor product multiplicities of Weyl modules for the classical groups in the stable range. Littlewood–Richardson (LR) coefficients form a special case. Klyachko connected eigenvalues of sums of Hermitian matrices to the saturated LR-cone and established defining linear inequalities. We prove analogues for the saturated NL-cone: a description by Extended Horn inequalities (as conjectured in part II of this series), where, using a result of King’s, this description is controlled by the saturated LR-cone and thereby recursive, just like the Horn inequalities; a minimal list of defining linear inequalities; an eigenvalue interpretation; and a factorization of Newell–Littlewood numbers, on the boundary.

We study the Macdonald intersection polynomials $\operatorname {I}_{\mu ^{(1)},\dots ,\mu ^{(k)}}[X;q,t]$, which are indexed by $k$-tuples of partitions $\mu ^{(1)},\dots ,\mu ^{(k)}$. These polynomials are conjectured to be equal to the bigraded Frobenius characteristic of the intersection of Garsia–Haiman modules, as proposed by the science fiction conjecture of Bergeron and Garsia. In this work, we establish the vanishing identity and the shape independence of the Macdonald intersection polynomials. Additionally, we unveil a remarkable connection between $\operatorname {I}_{\mu ^{(1)},\dots ,\mu ^{(k)}}$ and the character $\nabla e_{k-1}$ of diagonal coinvariant algebra by employing the plethystic formula for the Macdonald polynomials of Garsia, Haiman, and Tesler. Furthermore, we establish a connection between $\operatorname {I}_{\mu ^{(1)},\dots ,\mu ^{(k)}}$ and the shuffle formula $D_{k-1}[X;q,t]$, utilizing novel combinatorial tools such as the column exchange rule and the lightning bolt formula for Macdonald intersection polynomials. Notably, our findings provide a new proof for the shuffle theorem.

In this paper, we establish Newton–Maclaurin-type inequalities for functions arising from linear combinations of primitively symmetric polynomials. This generalization extends the classical Newton–Maclaurin inequality to a broader class of functions.

We find an Lascoux–Leclerc–Thibon (LLT)-type formula for a general power of the nabla operator of [BG99] applied to the Cauchy product for the modified Macdonald polynomials, and use it to deduce a new proof of the generalized shuffle theorem describing $\nabla^k e_n$ [HHL+05a, CM18, Mel21], and the formula for $(\nabla^k p_1^n,e_n)$ from [EH16, GH22] as corollaries. We give a direct proof of the theorem by verifying that the LLT expansion satisfies the defining properties of $\nabla^k$, such as triangularity in the dominance order, as well as a geometric proof based on a method for counting bundles on $\mathbb{P}^1$ due to the second author [Mel20]. These formulas are related to an affine paving of the type A unramified affine Springer fiber studied by Goresky, Kottwitz, and MacPherson in [GKM04], and also to Stanley’s chromatic symmetric functions.

In this study, we introduce multiple zeta functions with structures similar to those of symmetric functions such as the Schur P-, Schur Q-, symplectic and orthogonal functions in representation theory. Their basic properties, such as the domain of absolute convergence, are first considered. Then, by restricting ourselves to the truncated multiple zeta functions, we derive the Pfaffian expression of the Schur Q-multiple zeta functions, the sum formula for Schur P- and Schur Q-multiple zeta functions, the determinant expressions of symplectic and orthogonal Schur multiple zeta functions by making an assumption on variables. Finally, we generalize those to the quasi-symmetric functions.

Let W be a group endowed with a finite set S of generators. A representation $(V,\rho )$ of W is called a reflection representation of $(W,S)$ if $\rho (s)$ is a (generalized) reflection on V for each generator $s \in S$. In this article, we prove that for any irreducible reflection representation V, all the exterior powers $\bigwedge ^d V$, $d = 0, 1, \dots , \dim V$, are irreducible W-modules, and they are non-isomorphic to each other. This extends a theorem of R. Steinberg which is stated for Euclidean reflection groups. Moreover, we prove that the exterior powers (except for the 0th and the highest power) of two non-isomorphic reflection representations always give non-isomorphic W-modules. This allows us to construct numerous pairwise non-isomorphic irreducible representations for such groups, especially for Coxeter groups.

We examine bicoset digraphs and their natural properties from the point of view of symmetry. We then consider connected bicoset digraphs that are X-joins with collections of empty graphs, and show that their automorphism groups can be obtained from their natural irreducible quotients. We further show that such digraphs can be recognised from their connection sets.

The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ‘hubs’. We study the higher-order connectivity of such a network by investigating the topological properties of its clique complex. We concentrate on the Betti numbers, a sequence of topological invariants of the complex related to the numbers of holes (equivalently, repeated connections) of different dimensions. We prove that the expected Betti numbers grow sublinearly fast, with the trivial exceptions of those at dimensions 0 and 1. Our result also shows that preferential attachment graphs undergo infinitely many phase transitions within the parameter regime where the limiting degree distribution has an infinite variance. Regarding higher-order connectivity, our result shows that preferential attachment favors higher-order connectivity. We illustrate our theoretical results with simulations.

We give an explicit raising operator formula for the modified Macdonald polynomials $\tilde {H}_{\mu }(X;q,t)$, which follows from our recent formula for $\nabla $ on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions $\tilde {H}^{1,n}(X;q,t)$ that we call $1,n$-Macdonald polynomials, which reduce to a scalar multiple of $\tilde {H}_{\mu }(X;q,t)$ when $n=1$. We conjecture that the coefficients of $1,n$-Macdonald polynomials in terms of Schur functions belong to ${\mathbb N}[q,t]$, generalizing Macdonald positivity.

A pebble tree is an ordered tree where each node receives some colored pebbles, in such a way that each unary node receives at least one pebble, and each subtree has either one more or as many leaves as pebbles of each color. We show that the contraction poset on pebble trees is isomorphic to the face poset of a convex polytope called pebble tree polytope. Beside providing intriguing generalizations of the classical permutahedra and associahedra, our motivation is that the faces of the pebble tree polytopes provide realizations as convex polytopes of all assocoipahedra constructed by K. Poirier and T. Tradler only as polytopal complexes.

We study Schubert polynomials using geometry of infinite-dimensional flag varieties and degeneracy loci. Applications include Graham-positivity of coefficients appearing in equivariant coproduct formulas and expansions of back-stable and enriched Schubert polynomials. We also construct an embedding of the type C flag variety and study the corresponding pullback map on (equivariant) cohomology rings.