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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
$V$
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.
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.
Multi-compartment models described by systems of linear ordinary differential equations are considered. Catenary models are a particular class where the compartments are arranged in a chain. A unified methodology based on the Laplace transform is utilised to solve direct and inverse problems for multi-compartment models. Explicit formulas for the parameters in a catenary model are obtained in terms of the roots of elementary symmetric polynomials. A method to estimate parameters for a general multi-compartment model is also provided. Results of numerical simulations are presented to illustrate the effectiveness of the approach.
Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are strong Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of surfaces with orbifold points, establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their singularity categories, their Gorenstein dimensions and derived invariants such as the determinant of their q-Cartan matrices.
A noncomplete graph is
$2$
-distance-transitive if, for
$i \in \{1,2\}$
and for any two vertex pairs
$(u_1,v_1)$
and
$(u_2,v_2)$
with the same distance i in the graph, there exists an element of the graph automorphism group that maps
$(u_1,v_1)$
to
$(u_2,v_2)$
. This paper determines the family of
$2$
-distance-transitive Cayley graphs over dihedral groups, and it is shown that if the girth of such a graph is not
$4$
, then either it is a known
$2$
-arc-transitive graph or it is isomorphic to one of the following two graphs:
$ {\mathrm {K}}_{x[y]}$
, where
$x\geq 3,y\geq 2$
, and
$G(2,p,({p-1})/{4})$
, where p is a prime and
$p \equiv 1 \ (\operatorname {mod}\, 8)$
. Then, as an application of the above result, a complete classification is achieved of the family of
$2$
-geodesic-transitive Cayley graphs for dihedral groups.
Let $Q$ be an acyclic quiver and $w \geqslant 1$ be an integer. Let $\mathsf {C}_{-w}({\mathbf {k}} Q)$ be the $(-w)$-cluster category of ${\mathbf {k}} Q$. We show that there is a bijection between simple-minded collections in $\mathsf {D}^b({\mathbf {k}} Q)$ lying in a fundamental domain of $\mathsf {C}_{-w}({\mathbf {k}} Q)$ and $w$-simple-minded systems in $\mathsf {C}_{-w}({\mathbf {k}} Q)$. This generalises the same result of Iyama–Jin in the case that $Q$ is Dynkin. A key step in our proof is the observation that the heart $\mathsf {H}$ of a bounded t-structure in a Hom-finite, Krull–Schmidt, ${\mathbf {k}}$-linear saturated triangulated category $\mathsf {D}$ is functorially finite in $\mathsf {D}$ if and only if $\mathsf {H}$ has enough injectives and enough projectives. We then establish a bijection between $w$-simple-minded systems in $\mathsf {C}_{-w}({\mathbf {k}} Q)$ and positive $w$-noncrossing partitions of the corresponding Weyl group $W_Q$.
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 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 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.
We study the free metabelian group
$M(2,n)$
of prime power exponent n on two generators by means of invariants
$M(2,n)'\to \mathbb {Z}_n$
that we construct from colorings of the squares in the integer grid
$\mathbb {R} \times \mathbb {Z} \cup \mathbb {Z} \times \mathbb {R}$
. In particular, we improve bounds found by Newman for the order of
$M(2,2^k)$
. We study identities in
$M(2,n)$
, which give information about identities in the Burnside group
$B(2,n)$
and the restricted Burnside group
$R(2,n)$
.
Let G be a simple complex algebraic group, and let
$K \subset G$
be a reductive subgroup such that the coordinate ring of
$G/K$
is a multiplicity-free G-module. We consider the G-algebra structure of
$\mathbb C[G/K]$
and study the decomposition into irreducible summands of the product of irreducible G-submodules in
$\mathbb C[G/K]$
. When the spherical roots of
$G/K$
generate a root system of type
$\mathsf A$
, we propose a conjectural decomposition rule, which relies on a conjecture of Stanley on the multiplication of Jack symmetric functions. With the exception of one case, we show that the rule holds true whenever the root system generated by the spherical roots of
$G/K$
is a direct sum of subsystems of rank 1.
We compare crystal combinatorics of the level $2$ Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. We show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Furthermore, we find the supports of the unitary representations.