Let ${\mathscr {G}} $ be a special parahoric group scheme of twisted type over the ring of formal power series over $\mathbb {C}$, excluding the absolutely special case of $A^{(2)}_{2\ell }$. Using the methods and results of Zhu, we prove a duality theorem for general ${\mathscr {G}} $: there is a duality between the level one twisted affine Demazure modules and the function rings of certain torus fixed point subschemes in affine Schubert varieties for ${\mathscr {G}} $. Along the way, we also establish the duality theorem for $E_6$. As a consequence, we determine the smooth locus of any affine Schubert variety in the affine Grassmannian of ${\mathscr {G}} $. In particular, this confirms a conjecture of Haines and Richarz.

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 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 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.

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 give a complete combinatorial classification of the parabolic Verma modules in the principal block of the parabolic category $\mathcal{O}$ associated with a minimal or a maximal parabolic subalgebra of the special linear Lie algebra for which the answer to Kostant’s problem is positive.

We define oriented Temperley–Lieb algebras for Hermitian symmetric spaces. This allows us to explain the existence of closed combinatorial formulae for the Kazhdan–Lusztig polynomials for these spaces.

We extend the notion of ascent-compatibility from symmetric groups to all Coxeter groups, thereby providing a type-independent framework for constructing families of modules of $0$-Hecke algebras. We apply this framework in type B to give representation–theoretic interpretations of a number of noteworthy families of type-B quasisymmetric functions. Next, we construct modules of the type-B $0$-Hecke algebra corresponding to type-B analogs of Schur functions and introduce a type-B analog of Schur Q-functions; we prove that these shifted domino functions expand positively in the type-B peak functions. We define a type-B analog of the $0$-Hecke–Clifford algebra, and we use this to provide representation–theoretic interpretations for both the type-B peak functions and the shifted domino functions. We consider the modules of this algebra induced from type-B $0$-Hecke modules constructed via ascent-compatibility and prove a general formula, in terms of type-B peak functions, for the type-B quasisymmetric characteristics of the restrictions of these modules.

An identity that is reminiscent of the Littlewood identity plays a fundamental role in recent proofs of the facts that alternating sign triangles are equinumerous with totally symmetric self-complementary plane partitions and that alternating sign trapezoids are equinumerous with holey cyclically symmetric lozenge tilings of a hexagon. We establish a bounded version of a generalization of this identity. Further, we provide combinatorial interpretations of both sides of the identity. The ultimate goal would be to construct a combinatorial proof of this identity (possibly via an appropriate variant of the Robinson-Schensted-Knuth correspondence) and its unbounded version, as this would improve the understanding of the mysterious relation between alternating sign trapezoids and plane partition objects.

Assuming Stanley’s P-partitions conjecture holds, the regular Schur labeled skew shape posets are precisely the finite posets P with underlying set $\{1, 2, \ldots , |P|\}$ such that the P-partition generating function is symmetric and the set of linear extensions of P, denoted $\Sigma _L(P)$, is a left weak Bruhat interval in the symmetric group $\mathfrak {S}_{|P|}$. We describe the permutations in $\Sigma _L(P)$ in terms of reading words of standard Young tableaux when P is a regular Schur labeled skew shape poset, and classify $\Sigma _L(P)$’s up to descent-preserving isomorphism as P ranges over regular Schur labeled skew shape posets. The results obtained are then applied to classify the $0$-Hecke modules $\mathsf {M}_P$ associated with regular Schur labeled skew shape posets P up to isomorphism. Then we characterize regular Schur labeled skew shape posets as the finite posets P whose linear extensions form a dual plactic-closed subset of $\mathfrak {S}_{|P|}$. Using this characterization, we construct distinguished filtrations of $\mathsf {M}_P$ with respect to the Schur basis when P is a regular Schur labeled skew shape poset. Further issues concerned with the classification and decomposition of the $0$-Hecke modules $\mathsf {M}_P$ are also discussed.

Let ${\mathbf {x}}_{n \times n}$ be an $n \times n$ matrix of variables, and let ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ be the polynomial ring in these variables over a field ${\mathbb {F}}$. We study the ideal $I_n \subseteq {\mathbb {F}}[{\mathbf {x}}_{n \times n}]$ generated by all row and column variable sums and all products of two variables drawn from the same row or column. We show that the quotient ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ admits a standard monomial basis determined by Viennot’s shadow line avatar of the Schensted correspondence. As a corollary, the Hilbert series of ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ is the generating function of permutations in ${\mathfrak {S}}_n$ by the length of their longest increasing subsequence. Along the way, we describe a ‘shadow junta’ basis of the vector space of k-local permutation statistics. We also calculate the structure of ${\mathbb {F}}[{\mathbf {x}}_{n \times n}]/I_n$ as a graded ${\mathfrak {S}}_n \times {\mathfrak {S}}_n$-module.

We revisit Haiman’s conjecture on the relations between characters of Kazdhan–Lusztig basis elements of the Hecke algebra over $S_n$. The conjecture asserts that, for purposes of character evaluation, any Kazhdan–Lusztig basis element is reducible to a sum of the simplest possible ones (those associated to so-called codominant permutations). When the basis element is associated to a smooth permutation, we are able to give a geometric proof of this conjecture. On the other hand, if the permutation is singular, we provide a counterexample.

We study analogues of Kronecker coefficients for symmetric inverse semigroups, for dual symmetric inverse semigroups and for the inverse semigroups of bijections between subquotients of finite sets. In all cases, we reduce the problem of determination of such coefficients to some group-theoretic and combinatorial problems. For symmetric inverse semigroups, we provide an explicit formula in terms of the classical Kronecker and Littlewood–Richardson coefficients for symmetric groups.

In this paper, we give Pieri rules for skew dual immaculate functions and their recently discovered row-strict counterparts. We establish our rules using a right-action analogue of the skew Littlewood–Richardson rule for Hopf algebras of Lam–Lauve–Sottile. We also obtain Pieri rules for row-strict (dual) immaculate functions.