We prove quantitative bounds for the inverse theorem for Gowers uniformity norms $\mathsf {U}^5$ and $\mathsf {U}^6$ in $\mathbb {F}_2^n$. The proof starts from an earlier partial result of Gowers and the author which reduces the inverse problem to a study of algebraic properties of certain multilinear forms. The bulk of the work in this paper is a study of the relationship between the natural actions of $\operatorname {Sym}_4$ and $\operatorname {Sym}_5$ on the space of multilinear forms and the partition rank, using an algebraic version of regularity method. Along the way, we give a positive answer to a conjecture of Tidor about approximately symmetric multilinear forms in five variables, which is known to be false in the case of four variables. Finally, we discuss the possible generalization of the argument for $\mathsf {U}^k$ norms.

A result of Corfield, Sati, and Schreiber asserts that $\mathfrak {gl}_{n}$-weight systems associated with the defining representation are quantum states. In this short note, we extend this result to all $\mathfrak {gl}_{n}$-weight systems corresponding to labeling by symmetric and exterior powers of the defining representation.

We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose x and y intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $\operatorname {\mathrm {GL}}_{l}$ characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials.

We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $\Delta _{h_l}\Delta ' _{e_k} e_{n}$, where $\Delta ' _{e_k}$ and $\Delta _{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $\operatorname {\mathrm {GL}}_m$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.

A subset Y of the general linear group $\text{GL}(n,q)$ is called t-intersecting if $\text{rk}(x-y)\le n-t$ for all $x,y\in Y$, or equivalently x and y agree pointwise on a t-dimensional subspace of $\mathbb{F}_q^n$ for all $x,y\in Y$. We show that, if n is sufficiently large compared to t, the size of every such t-intersecting set is at most that of the stabiliser of a basis of a t-dimensional subspace of $\mathbb{F}_q^n$. In case of equality, the characteristic vector of Y is a linear combination of the characteristic vectors of the cosets of these stabilisers. We also give similar results for subsets of $\text{GL}(n,q)$ that intersect not necessarily pointwise in t-dimensional subspaces of $\mathbb{F}_q^n$ and for cross-intersecting subsets of $\text{GL}(n,q)$. These results may be viewed as variants of the classical Erdős–Ko–Rado Theorem in extremal set theory and are q-analogs of corresponding results known for the symmetric group. Our methods are based on eigenvalue techniques to estimate the size of the largest independent sets in graphs and crucially involve the representation theory of $\text{GL}(n,q)$.

A subset R of the vertex set of a graph $\Gamma $ is said to be $(\kappa ,\tau )$-regular if R induces a $\kappa $-regular subgraph and every vertex outside R is adjacent to exactly $\tau $ vertices in R. In particular, if R is a $(\kappa ,\tau )$-regular set of some Cayley graph on a finite group G, then R is called a $(\kappa ,\tau )$-regular set of G. Let H be a nontrivial normal subgroup of G, and $\kappa $ and $\tau $ a pair of integers satisfying $0\leq \kappa \leq |H|-1$, $1\leq \tau \leq |H|$ and $\gcd (2,|H|-1)\mid \kappa $. It is proved that (i) if $\tau $ is even, then H is a $(\kappa ,\tau )$-regular set of G; (ii) if $\tau $ is odd, then H is a $(\kappa ,\tau )$-regular set of G if and only if it is a $(0,1)$-regular set of G.

We study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let $R_n$ denote the ring of polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables:

1. (1) The quasisymmetric polynomials in $R_n$ form a commutative subalgebra of $R_n$.

2. (2) There is a basis of the quotient of $R_n$ by the ideal $I_n$ generated by the quasisymmetric polynomials in $R_n$ that is indexed by ballot sequences. The Hilbert series of the quotient is given by

   $$ \begin{align*}\text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,\end{align*} $$
   where $f^{(n-k,k)}$ is the number of standard tableaux of shape $(n-k,k)$.

3. (3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition.

Let G be a finite group. Let $H, K$ be subgroups of G and $H \backslash G / K$ the double coset space. If Q is a probability on G which is constant on conjugacy classes ($Q(s^{-1} t s) = Q(t)$), then the random walk driven by Q on G projects to a Markov chain on $H \backslash G /K$. This allows analysis of the lumped chain using the representation theory of G. Examples include coagulation-fragmentation processes and natural Markov chains on contingency tables. Our main example projects the random transvections walk on $GL_n(q)$ onto a Markov chain on $S_n$ via the Bruhat decomposition. The chain on $S_n$ has a Mallows stationary distribution and interesting mixing time behavior. The projection illuminates the combinatorics of Gaussian elimination. Along the way, we give a representation of the sum of transvections in the Hecke algebra of double cosets, which describes the Markov chain as a mixture of Metropolis chains. Some extensions and examples of double coset Markov chains with G a compact group are discussed.

The superspace ring $\Omega _n$ is a rank n polynomial ring tensored with a rank n exterior algebra. Using an extension of the Vandermonde determinant to $\Omega _n$, the authors previously defined a family of doubly graded quotients ${\mathbb {W}}_{n,k}$ of $\Omega _n$, which carry an action of the symmetric group ${\mathfrak {S}}_n$ and satisfy a bigraded version of Poincaré Duality. In this paper, we examine the duality modules ${\mathbb {W}}_{n,k}$ in greater detail. We describe a monomial basis of ${\mathbb {W}}_{n,k}$ and give combinatorial formulas for its bigraded Hilbert and Frobenius series. These formulas involve new combinatorial objects called ordered set superpartitions. These are ordered set partitions $(B_1 \mid \cdots \mid B_k)$ of $\{1,\dots ,n\}$ in which the nonminimal elements of any block $B_i$ may be barred or unbarred.

We prove new mixing rate estimates for the random walks on homogeneous spaces determined by a probability distribution on a finite group $G$. We introduce the switched random walk determined by a finite set of probability distributions on $G$, prove that its long-term behaviour is determined by the Fourier joint spectral radius of the distributions, and give Hermitian sum-of-squares algorithms for the effective estimation of this quantity.

We demonstrate that two supersoluble complements of an abelian base in a finite split extension are conjugate if and only if, for each prime $p$, a Sylow $p$-subgroup of one complement is conjugate to a Sylow $p$-subgroup of the other. As a corollary, we find that any two supersoluble complements of an abelian subgroup $N$ in a finite split extension $G$ are conjugate if and only if, for each prime $p$, there exists a Sylow $p$-subgroup $S$ of $G$ such that any two complements of $S\cap N$ in $S$ are conjugate in $G$. In particular, restricting to supersoluble groups allows us to ease D. G. Higman's stipulation that the complements of $S\cap N$ in $S$ be conjugate within $S$. We then consider group actions and obtain a fixed point result for non-coprime actions analogous to Glauberman's lemma.

Let n be a nonnegative integer. For each composition $\alpha $ of n, Berg, Bergeron, Saliola, Serrano and Zabrocki introduced a cyclic indecomposable $H_n(0)$-module $\mathcal {V}_{\alpha }$ with a dual immaculate quasisymmetric function as the image of the quasisymmetric characteristic. In this paper, we study $\mathcal {V}_{\alpha }$s from the homological viewpoint. To be precise, we construct a minimal projective presentation of $\mathcal {V}_{\alpha }$ and a minimal injective presentation of $\mathcal {V}_{\alpha }$ as well. Using them, we compute $\mathrm {Ext}^1_{H_n(0)}(\mathcal {V}_{\alpha }, \mathbf {F}_{\beta })$ and $\mathrm {Ext}^1_{H_n(0)}( \mathbf {F}_{\beta }, \mathcal {V}_{\alpha })$, where $\mathbf {F}_{\beta }$ is the simple $H_n(0)$-module attached to a composition $\beta $ of n. We also compute $\mathrm {Ext}_{H_n(0)}^i(\mathcal {V}_{\alpha },\mathcal {V}_{\beta })$ when $i=0,1$ and $\beta \le _l \alpha $, where $\le _l$ represents the lexicographic order on compositions.

Let $V$ be a finite-dimensional vector space over $\mathbb{F}_p$. We say that a multilinear form $\alpha \colon V^k \to \mathbb{F}_p$ in $k$ variables is $d$-approximately symmetric if the partition rank of difference $\alpha (x_1, \ldots, x_k) - \alpha (x_{\pi (1)}, \ldots, x_{\pi (k)})$ is at most $d$ for every permutation $\pi \in \textrm{Sym}_k$. In a work concerning the inverse theorem for the Gowers uniformity $\|\!\cdot\! \|_{\mathsf{U}^4}$ norm in the case of low characteristic, Tidor conjectured that any $d$-approximately symmetric multilinear form $\alpha \colon V^k \to \mathbb{F}_p$ differs from a symmetric multilinear form by a multilinear form of partition rank at most $O_{p,k,d}(1)$ and proved this conjecture in the case of trilinear forms. In this paper, somewhat surprisingly, we show that this conjecture is false. In fact, we show that approximately symmetric forms can be quite far from the symmetric ones, by constructing a multilinear form $\alpha \colon \mathbb{F}_2^n \times \mathbb{F}_2^n \times \mathbb{F}_2^n \times \mathbb{F}_2^n \to \mathbb{F}_2$ which is 3-approximately symmetric, while the difference between $\alpha$ and any symmetric multilinear form is of partition rank at least $\Omega (\sqrt [3]{n})$.

Motivated by a new conjecture on the behavior of bricks, we start a systematic study of minimal $\tau $-tilting infinite (min-$\tau $-infinite, for short) algebras. In particular, we treat min-$\tau $-infinite algebras as a modern counterpart of minimal representation-infinite algebras and show some of the fundamental similarities and differences between these families. We then relate our studies to the classical tilting theory and observe that this modern approach can provide fresh impetus to the study of some old problems. We further show that in order to verify the conjecture, it is sufficient to treat those min-$\tau $-infinite algebras where almost all bricks are faithful. Finally, we also prove that minimal extending bricks have open orbits, and consequently obtain a simple proof of the brick analogue of the first Brauer–Thrall conjecture, recently shown by Schroll and Treffinger using some different techniques.

We study multivariate polynomials over ‘structured’ grids. Firstly, we propose an interpretation as to what it means for a finite subset of a field to be structured; we do so by means of a numerical parameter, the nullity. We then extend several results – notably, the Combinatorial Nullstellensatz and the Coefficient Theorem – to polynomials over structured grids. The main point is that the structure of a grid allows the degree constraints on polynomials to be relaxed.

We study the compatibility of the action of the DAHA of type GL with two inverse systems of polynomial rings obtained from the standard Laurent polynomial representations. In both cases, the crucial analysis is that of the compatibility of the action of the Cherednik operators. Each case leads to a representation of a limit structure (the +/– stable limit DAHA) on a space of almost symmetric polynomials in infinitely many variables (the standard representation). As an application, we show that the defining representation of the double Dyck path algebra arises from the standard representation of the +stable limit DAHA.

For a simple bipartite graph G, we give an upper bound for the regularity of powers of the edge ideal $I(G)$ in terms of its vertex domination number. Consequently, we explicitly compute the regularity of powers of the edge ideal of a bipartite Kneser graph. Further, we compute the induced matching number of a bipartite Kneser graph.

In this note, we correct an oversight regarding the modules from Definition 4.2 and proof of Lemma 5.12 in Baur et al. (Nayoga Math. J., 2020, 240, 322–354). In particular, we give a correct construction of an indecomposable rank $2$ module $\operatorname {\mathbb {L}}\nolimits (I,J)$, with the rank 1 layers I and J tightly $3$-interlacing, and we give a correct proof of Lemma 5.12.

We construct coloured lattice models whose partition functions represent symplectic and odd orthogonal Demazure characters and atoms. We show that our lattice models are not solvable, but we are able to show the existence of sufficiently many solutions of the Yang–Baxter equation that allow us to compute functional equations for the corresponding partition functions. From these functional equations, we determine that the partition function of our models are the Demazure atoms and characters for the symplectic and odd orthogonal Lie groups. We coin our lattice models as quasi-solvable. We use the natural bijection of admissible states in our models with Proctor patterns to give a right key algorithm for reverse King tableaux and Sundaram tableaux.

Let K be an infinite field of characteristic $p>0$ and let $\lambda, \mu$ be partitions, where $\mu$ has two parts. We find sufficient arithmetic conditions on $p, \lambda, \mu$ for the existence of a nonzero homomorphism $\Delta(\lambda) \to \Delta (\mu)$ of Weyl modules for the general linear group $GL_n(K)$. Also, for each p we find sufficient conditions so that the corresponding homomorphism spaces have dimension at least 2.