We establish a canonical Bernstein--Zelevinsky filtration for Casselman--Wallach representations that is analogous to the $p$-adic case. In addition, we outline an approach to Casselman's comparison conjecture and prove it for general linear groups, as well as for quasi-split even orthogonal groups in some special cases. We also give some applications of the Bernstein--Zelevinsky filtration, such as to the study of highest derivatives and the indecomposability of mirabolic restrictions.

Given a finite group $G$, we prove that there exist signs $\varepsilon\in\{\pm1\}^G$ such that $$\left\| \sum_{g\in G} \varepsilon_g\rho(g) \right\|\leq C\, \sqrt{|G|},$$ where $\rho$ is the left regular representation of $G$, and $C$ is a universal constant. This special case of the Matrix Spencer conjecture was posed in [BKMZ24], where it was established for simple groups.

We establish the Kazhdan isomorphism for the $n$-fold metaplectic cover of the reductive group $\mathrm{SL}_2$ over two sufficiently close non-archimedean local fields $F$ and $F'$, both of which have residue characteristic coprime to $n$ and contain all distinct $n$th roots of unity.

Let $L(\lambda)$ be a simple highest weight module of a classical Lie algebra $\mathfrak{g}$ with highest weight $\lambda-\rho$, where $\rho$ is half the sum of positive roots. Joseph proved that the associated variety of the annihilator ideal of $L(\lambda)$ (also called the annihilator variety) is the Zariski closure of a nilpotent orbit in $\mathfrak{g}^*$. Recently, Bai--Ma--Wang introduced a partition algorithm to describe this corresponding nilpotent orbit for a given highest weight module $L(\lambda)$. In this paper, we present a new direct proof of Bai--Ma--Wang's partition algorithm using Sommers duality.

In this work, we settle a problem that dates back to the early 2000s. We provide a complete characterization of special matchings of arbitrary Bruhat intervals in Coxeter groups of type $A$ and apply this result to prove a conjecture of Brenti from 2003 concerning the computation of Kazhdan-Lusztig $R$-polynomials via special matchings. This yields new evidence in support of the Combinatorial Invariance Conjecture.

Let $A$ be a finite-dimensional basic algebra over an algebraically closed field $K$, $T$ a finitely generated $\tau$-tilting right $A$-module and $B={\rm End}_A T$. Denote by ${\rm Fac}T$ the subcategory of finitely generated right $A$-modules generated by $T$. We define the depth relative to $T$ and the delooping level relative to $T$ and show that the finitistic dimension of the opposite algebra of $B$ is bounded by the depth of $\textup{Fac}T$ relative to $T$ and the delooping level of $\textup{Fac}T$ relative to $T$. We give applications to the finitistic dimension conjecture. More precisely, we show that if $A$ is a minimal representation infinite algebra or an algebra of finite representation type, then the finitistic dimension of $B^{op}$ is finite.

If A is a symmetric algebra, then Hochschild homology of the dg enhancement of the singularity category of A agrees with singular Hochschild homology of A. For a basic finite dimensional k algebra A, the Cartan matrix of A is symmetric if and only if the k dual of the mixed complex of the dg enhancement of its singularity category is isomorphic to its shift by -1. We provide two counterexamples to show that neither result holds for general Frobenius algebras.

In this paper we develop a combinatorial algorithm to compute the Gelfand--Kirillov (GK) dimension of simple highest weight modules for basic classical Lie superalgebras. Building upon the results for classical Lie algebras via Lusztig's {\bf a}-function and the Robinson--Schensted (RS) insertion algorithm, we extend these techniques to the super setting, providing explicit formulas for types $\mathfrak{sl}(m|n)$ and $\mathfrak{osp}(2|2n)$. Our results show that the GK dimension of a simple highest weight module is determined entirely by the even part of the Lie superalgebras.

We prove the Harish-Chandra isomorphism theorem for supersymmetric spaces, describing the polynomial algebra of eigenvalues of invariant differential operators. The polynomials obtained satisfy novel invariance conditions, which remain somewhat mysterious. We also prove the Harish-Chandra isomorphism for ghost distributions, which satisfy a `square root' of the invariance conditions coming from invariant differential operators. All proofs are algebraic, and rely on a rank-one reduction argument and the Chevalley restriction theorem.

The Terwilliger algebras of association schemes over an arbitrary field $\mathbb{F}$ were called the Terwilliger $\mathbb{F}$-algebras of association schemes in [10]. In [7], He and Jiang studied the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. In this paper, we continue studying the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. We get all block idempotents of the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. We get the $\mathbb{F}$-dimensions, the centers, the Jacobson radicals of the block algebras of the Terwilliger $\mathbb{F}$-algebras of factorial association schemes.

In this paper, we develop a quiver approach to coquasi-Hopf algebras with the dual Chevalley property. We introduce a modified generalized path coalgebra $\Bbbk(\mathrm{Q},\mathcal{S})$ associated with a given quiver $\mathrm{Q}$ and a collection of simple coalgebras $\mathcal{S}=\{C_i\mid i\in \mathrm{Q}_0\}$ indexed by the vertices of $\mathrm{Q}$, such that its link quiver coincides with $\mathrm{Q}$. We prove that such a coalgebra admits a graded coquasi-Hopf algebra structure with the dual Chevalley property if and only if $\mathrm{Q}$ is a generalized Hopf quiver and $\bigoplus_{i\in \mathrm{Q}_0}C_i$ forms a cosemisimple coquasi-Hopf algebra. Moreover, we provide a classification of these coquasi-Hopf algebra structures. We then study the link-indecomposable components of a coquasi-Hopf algebra with the dual Chevalley property, and give the generalized dual Gabriel's theorem for such coquasi-Hopf algebras. As an application, we apply the quiver method to classify finite integral tensor categories with the Chevalley property of finite representation type. We also give structural characterizations of coradically graded coquasi-Hopf algebras of tame corepresentation type. Furthermore, we investigate finite braided integral tensor categories with the Chevalley property via the quiver approach.

Let $G$ be a finite group and $p$ a prime. It is interesting to determine when two ordinary irreducible representations of $G$ have the same $p$-modular reduction; this is the same as saying that the corresponding rows of the decomposition matrix are equal, or that the characters of the two representations agree on $p$-regular conjugacy classes. In fact we consider the more general problem of asking when two rows of the decomposition matrix are proportional. In the case where $G$ is a double cover of the alternating or symmetric group, this problem has been solved except when $p=3$. Here we resolve the missing case for spin characters (i.e. characters which are not lifted from the covered group), which completely solves the problem for the double cover of the symmetric group. There are surprising parallels to our solution to the corresponding problem for $p=2$.

We classify when a spin and a non-spin irreducible character of the alternating group have proportional 2-modular reductions. Equivalently, we classify when such a pair of characters are proportional on elements of odd order.

We consider the $SO(d)$ and $O(d)$ invariant sectors of the bosonic $d$-matrix harmonic oscillator with $U(N)$ gauge symmetry. The micro-canonical degeneracy $\mathcal{Z}( N, d, k )$ for fixed energy $k$ is expressed as a pairing between an $N$-dependent vector and a $d$-dependent vector in the space of partitions of the integer $k$. This pairing formula is derived by counting invariant words in multi-matrix variables $X^i_{j,a}$, using properties of Clebsch-Gordan multiplicities (Kronecker coefficients) for the symmetric group $S_k$, Schur-Weyl duality and harmonic analysis on the homogeneous space $U(d)/SO(d)$. Analytic formulae for large $N$ and $k$ with $ k \le N $ are obtained using group integrals over $U(N)$ and $SO(d)$ (or $ O(d)$). The micro-canonical heat capacity in this regime is negative and turns positive, at a critical value $k_{\rm crit}$, due to finite $N$ modifications to the counting, thus forming what we denote as a characteristic caloric fold in the $ E $ versus $T$ curve. Data from the pairing formula is well fitted by $k_{\rm crit} \sim { N^2 \over 4 }$ for small values of $d$. A derivation of this large $N$ formula is given using a matrix model approximation and semi-classical analysis of the eigenvalue density. The large $N,d$ limit of the degeneracies reveals a key role for ribbon graph combinatorics. The caloric fold is also notably a property of black hole thermodynamics in anti-de-Sitter spaces. We propose the spherically symmetric \(SO(d)\) and \(O(d)\) invariant sectors of \(d\)-matrix quantum mechanics as tractable matrix systems for capturing key features of dual descriptions of black-hole thermodynamics.

We establish the existence of abelian envelopes for interpolation categories of wreath product groups $G\wr S_n$, for a fixed finite group $G$ with the symmetric groups $S_n$, for $n\ge0$. Our approach consists of showing directly via essentially combinatorial methods that certain generalized restriction functors admit adjoints.

While the uncrowding algorithm on set-valued tableaux has long been instrumental in proving the Schur positivity of stable symmetric Grothendieck polynomials, lattice models have emerged as a modern framework for investigating symmetric functions, in particular symmetric Grothendieck polynomials. In this work, we synthesize these combinatorial and lattice-theoretic approaches by defining both the Robinson--Schensted--Knuth (RSK) correspondence and the uncrowding operation directly on a 5-vertex model of Motegi and Sakai and its subsequent reinterpretation by Buciumas, Scrimshaw, and Weber. Our lattice-based RSK formulation yields a powerful new result: the direct construction of the associated crystal structure on the states of the 5-vertex model.

We survey theory developed over the past 10 years of semirings which need not be additively cancellative. The main features are a specified ``null ideal'' $\mcA_0$ of a semiring $\mcA,$ taking the place of a zero element, and a ``surpassing relation,'' taking the place of equality, which permit generalizations of the classical algebraic theory to polynomials and their roots, algebraic geometry, matrices, linear algebra, varieties, categories, and module theory. The ``pair'' $(\mcA,\mcA_0)$ is studied along the lines of universal algebra.

We show how monoidal adjunctions can be used to prove the existence of monoidal abelian envelopes of pseudo-tensor categories, in particular, those admitting a combinatorial description with certain properties. We derive concrete general criteria, which we then demonstrate by giving relatively simple combinatorial proofs of the existence of new abelian envelopes for interpolation categories of the hyperoctahedral and of the modified symmetric groups.

Let $R$ be a noetherian algebra over a Cohen--Macaulay ring $S$ admitting a canonical module $\omega$, and assume that $R$ is maximal Cohen--Macaulay over $S$. We prove that the category of finitely generated Gorenstein projective $R$-modules coincides with the left $\mathrm Ext$-orthogonal class of the thick subcategory generated by $R$ and ${\mathrm Hom}_S(R,\omega)$. As an application, finitely generated Gorenstein projective $R$-modules form the left half of a hereditary cotorsion pair. In the case of Cohen--Macaulay local rings, this yields an affirmative answer to a question of R. Takahashi. We further characterize when $R$ is left weakly Gorenstein. Finally, we prove that a Cohen--Macaulay local ring is Gorenstein if and only if the right $\mathrm Ext$-orthogonal class of finitely generated Gorenstein projective modules coincides with the category of finitely generated modules of finite projective dimension.

We consider the Hitchin fibration on the moduli stack of Higgs bundles with arbitrary reductive structure group, and study its singular locus using the centraliser of the Higgs field. We restrict to the case where the Higgs field has constant centraliser dimension, and describe a non-abelian structure on the corresponding locus in the moduli stack. On a class of components of this locus, we construct a factorisation of the Hitchin map through an abelianised fibration, and describe the abelianised fibres with a generalisation of the cameral data of Donagi and Gaitsgory. We apply our results to Hitchin fibrations for real groups, and we also determine a connection between the geometry of the singular Hitchin fibration and the representation theory of the Lie algebra via the orbit method.

Using the spinless Haldane model, we study the witness-filtered Berry curvature, quantum geometric tensor, and quantum Fisher information on the gapped strata of the parameter space and evaluate them through the Fukui-Hatsugai-Suzuki discretization. The filtered quantities isolate the part of the geometric response carried by sublattice coherence: they suppress contributions from regions where the occupied Bloch state is locally A/B-separable and emphasize regions where curvature and coherence coexist. We derive exact lattice identities, reconstruction formulas for the curvature-weighted coherence, and bounds relating the filtered quantum geometric tensor and quantum Fisher information to single-particle mode entanglement. Across the gap-closing stratum, the quantized response changes admit a natural description in terms of Hecke modifications. We elicit a corresponding Langlands viewpoint -- not as a full correspondence, but as an organizational principle and as the mathematical shadow of these physical geometric constructions.

We describe a framework for encoding cluster combinatorics using categorical methods. We give a definition of an abstract cluster structure, which captures the essence of cluster mutation at a tropical level and show that cluster algebras, cluster varieties, cluster categories and surface models all have associated abstract cluster structures. For the first two classes, we also show that they can be constructed from abstract cluster structures. By defining a suitable notion of morphism of abstract cluster structures, we introduce a category of these and show that it has several desirable properties, such as initial and terminal objects and finite products and coproducts. We also prove that rooted cluster morphisms of cluster algebras give rise to morphisms of the associated abstract cluster structures, so that our framework includes a version of the extant category of cluster algebras. We can do more, however, because we can relate different types of representation of abstract cluster structures (cluster algebra, varieties, categories) directly via morphisms of their associated abstract cluster structures, even though no direct map from e.g. a cluster category to the associated cluster algebra is possible. In fact, we do much of the above in the setting of abstract quantum cluster structures, with some analysis of the difference between the category of these and that of the unquantized version. In order to show the relationship between abstract quantum cluster structures and quantum cluster algebras, we reformulate the usual construction of the latter in a way that is more amenable to our purposes and which we expect will be of independent interest and use.

We develop theory and examples of monoidal functors on tensor categories in positive characteristic that generalise the Frobenius functor from \cite{Os, EOf, Tann}. The latter has proved to be a powerful tool in the ongoing classification of tensor categories of moderate growth, and we demonstrate the similar potential of the generalisations. More explicitly, we describe a new construction of the generalised Verlinde categories $Ver_{p^n}$ in terms of representation categories of elementary abelian $p$-groups. This leads to families of functors relating to $Ver_{p^n}$ that we conjecture, and partially show, to exhibit the characteristic properties of the Frobenius functor relating to $Ver_p$. In particular, we conjecture some of these functors to detect categories that fibre over $Ver_{p^n}$.

Given a monoidal adjunction, we show that the right adjoint induces a braided lax monoidal functor between the corresponding Drinfeld centers provided that certain natural transformations, called projection formula morphisms, are invertible. We investigate these induced functors on Drinfeld centers in more detail for the monoidal adjunction of restriction and (co-)induction along morphisms of Hopf algebras. The resulting functors are applied to examples related to affine algebraic groups, quantum groups at roots of unity, and Radford--Majid biproducts of Hopf algebras. Moreover, we use the projection formula morphisms to prove a characterization theorem for monoidal Kleisli adjunctions and a crude monoidal monadicity theorem. The functor on Drinfeld centers induced by the Eilenberg--Moore adjunction is given in terms of local modules over commutative central monoids.