Papers

We establish a multiplicity-one refinement of the non-archimedean type-II Howe correspondence: whenever the small theta lift is non-zero, it occurs in the big theta lift with Jordan-Hölder multiplicity one. Furthermore, we apply this result alongside the Godement-Jacquet zeta integral to compute big theta lift for Arthur-type representations. Based on these computations, we also propose a conjecture regarding the irreducibility of the big theta lift.

Tilting mutation provides a natural way to construct derived-equivalent algebras by replacing an indecomposable summand of a tilting object. Holm, Schröer and Zimmermann~\cite{HSZ01} (see also the appendix of~\cite{AA08}) have described combinatorial transformations of the quiver and relations of a gentle algebra that yield derived equivalences via tilting complexes. In this paper, we develop a self-contained module-theoretic framework for tilting mutation via generalized $\mathrm{BB}$-tilting modules. For a gentle algebra $A=\mathbb{k} Q/\langle I\rangle$ and a vertex $k\in Q_0$, we first give a necessary and sufficient condition, expressed in terms of the arrows and relations incident with $k$, for the corresponding minimal left approximation of $P(k)$ to yield a tilting mutation. This criterion applies uniformly to vertices with or without loops. When the mutation exists, we determine the irreducible morphisms between the indecomposable summands of the mutated tilting module and use them to construct explicitly the Gabriel quiver and defining relations of its endomorphism algebra. In particular, we also provide a combinatorial mutation $(Q,I)\mapsto(Q',I')$ of gentle pairs such that $\mu_k^+(A)\cong \mathbb{k} Q'/\langle I'\rangle,$ so the resulting algebra is again gentle and derived equivalent to $A$. Finally, we formulate the dual cotilting mutation and its combinatorial description via opposite gentle pairs.

We study quiver varieties associated with orthogonal quivers and determine the Kleinian surface singularities arising from affine Dynkin types. In the classical setting, affine Dynkin quivers play a distinguished role through Kronheimer's construction of Kleinian singularities and its reformulation by Nakajima in terms of quiver varieties. Using a graphical calculus in the preprojective algebra of the quiver, we determine which Kleinian surface singularities arise from orthogonal quivers whose underlying graph is an affine Dynkin diagram.

This paper studies orthogonal model structures, i.e., model structures such that a lifting in the Lifting axiom is unique. Cofibrant (fibrant) objects are defined without initial (terminal) objects. For an orthogonal model structure on a category with enough cofibrant objects and fibrant objects, it is proved that the homotopy category is equivalent to the full subcategory of cofibrant-fibrant objects. TTF model structures, bi-reflective model structures, and torsion model structures, are introduced. They are all orthogonal. One to one correspondences between TTF model structures and TTF triples in an abelian category, bi-reflective model structures and bi-reflective pairs in any category, and torsion model structures and twin torsion pairs in an abelian category, are established in a constructive way. Torsion model structures on a poset, on the category of $G$-sets, and on the category of topological groups, are also constructed. The homotopy categories of all these model structures are computed. In particular, the homotopy category of a TTF model structure is an abelian category.

To a quiver of finite type one can associate two abelian categories: the category of finite dimensional modules over the corresponding affine quiver Hecke algebra and the Coulomb category of Koszul-perverse coherent sheaves. In this paper we define an exact functor from the former to the latter. This functor sends a simple to a simple or zero and is (almost) essentially surjective on simples. This implies that simple Koszul-perverse sheaves categorify the dual canonical basis.

We establish an explicit formula for the Kostka–Foulkes polynomial of type $C_n$ with unequal parameters at fundamental weights. As an application, we deduce an explicit Lusztig–Kato formula for the Hecke algebra of the special (but not hyperspecial) maximal compact subgroup of a quasi-split but non-split special orthogonal group over a $p$-adic field.

We study the singularities of orbit closures in loop symmetric varieties, extending our previous work https://arxiv.org/abs/2310.20006 to the twisted setting. We prove that the IC complexes of orbit closures are pointwise pure and satisfy a parity-vanishing property. We apply these geometric results to the study of twisted affine Lusztig–Vogan modules, establishing foundational results, including positivity properties of the twisted affine Kazhdan–Lusztig–Vogan polynomials. Along the way, we construct conical symplectic transversal slices in loop symmetric spaces. We provide applications to Langlands duality for real groups.

Let $\mathcal{E}$ be an essentially small idempotent-complete exact category with enough projective objects. We develop derived delooping levels in $\mathcal{E}$. We apply this framework to a basic support $\tau$-tilting module $T$. We then define the derived delooping level relative to $T$ and compare it with the ordinary relative delooping level and the derived delooping level of $B=\operatorname{End}_A(T)$. We show that the big finitistic dimension of $B^{\mathrm{op}}$ is bounded above by the derived delooping level of $\operatorname{Fac}T$ relative to $T$. Moreover, we prove that a self-orthogonal $\tau$-tilting module $T$ is a $1$-tilting module whenever $2\text{-}\ddell_B(DT)$ is finite.

Let \(\cB\) be an abelian category with enough projective and injective objects, and let $ \cA=\cB\ltimes_\eta\sF $ be an \(\eta\)-extension induced by a right exact endofunctor \(\sF\). We introduce uniform image-nilpotence for \(\eta\)-extensions and characterize it in terms of the nilpotence of the associative natural transformation \(\eta:\sF^2\to\sF\). Using the associated \(\eta\)-Loewy filtration, we prove a lifting theorem for right \(n\)-cotorsion pairs over image-nilpotent \(\eta\)-extensions and establish a corresponding completeness theorem. As applications, our results recover and unify related constructions for split nilpotent ring extensions, comma categories, formal triangular matrix rings, Morita context rings and classical trivial extensions.

We introduce a combinatorial criterion for the projectivity of standard modules over generalized Reedy categories, formulated in terms of a finite bipartite graph whose normalized weightings determine normalization idempotents and hence projective splittings of standard modules. We apply this framework to recover several classical normalization constructions and Dold–Kan–type equivalences, and to obtain new Dold–Kan–type equivalences for categories of finite-dimensional affine, semilinear, and semiaffine spaces over a finite field. These equivalences also yield explicit descriptions of uniformly continuous representations of the corresponding infinite transformation monoids. Together with our previous work on Hom-spaces between standard modules, the results provide a combinatorial machinery, based mainly on graph theory and linear algebra, for establishing Dold–Kan–type equivalences for generalized Reedy categories.

Let $ G $ be a non-compact connected reductive real Lie group and $ P $ its parabolic subgroup. We give a list of equivalent conditions for $ P $ to have an abelian unipotent radical. These conditions are related to skew multiplicity free and multiplicity free actions, and using the theory of Clifford algebra and the cubic Dirac operator we deduce formulas for the de Rham cohomology of compact symmetric spaces $ G/P $.

Homogeneous coordinate rings of Grassmannians are among the most fundamental instances of cluster algebras arising in “nature.” Despite their importance, however, the cluster variables of these cluster algebras are not well-understood, except in special cases. For the Grassmannian of 3-planes in n-space, Fomin and Pylyavskyy conjectured [FP16] that the cluster and frozen variables are exactly the indecomposable, non-elliptic web invariants which are arborizable. They proved that in particular, if an SL3 tensor diagram is a planar tree, its web invariant is a cluster or frozen variable. In this article, we give a new proof of their result for Gr(3, n) planar tree webs, and we generalize the result to Gr(4, n): if an SL4 tensor diagram for Gr(4, n) is a planar tree web, its web invariant is a cluster or frozen variable. Our proof uses some of the cluster quasihomomorphisms from [EZLP+23] and [EZPT+25] (namely, upper promotion and spurion promotion), as well as a new 6-leg pattern map.

The aim of the paper is to study Euclidean and l-finite Euclidean modules over a ring, submodules of their finite direct sums, reduced and strong reduced echelon forms of matrices with coefficients in Euclidean modules and bimodules. The concept of Euclidean module is a module-theoretic analogue of the concept of Euclidean ring. The class of (not necessarily commutative) Euclidean rings is a nice but `very small' class of rings that have arithmetic-like properties. On the contrary, surprisingly, the class of Euclidean modules is a `large class' of modules. The Euclidean modules exist for many constructions of rings that are defined over arbitrary rings. For example, for every skew polynomial ring $A=D[x;\s, \d]$ with coefficients in an arbitrary ring $D$, the generalized Weyl algebra $D[x,y;\s, a]$, the Weyl algebra $A_1=K[x][\der; \frac{d}{dx}]$, $U(\sl2)$ and many classical algebras, there exist Euclidean modules.

The interval $[1,c]_T$ between the identity element and a Coxeter element $c$ in the absolute order on a Coxeter group $W$ is a generalization of the poset of noncrossing partitions arising when $W$ is the symmetric group. When $W$ is finite, this poset is always a lattice, and it is natural to associate to every element $w\in [1,c]_T$ its moved space $\mathsf{Mov}(w)=\mathrm{Im}(w - \mathrm{Id}_V)$ in the geometric representation $V$ of $W$. It has dimension equal to the reflection length $\ell_T(w)$ of $w$, and gives a realization of $[1,c]_T$ inside the lattice of subspaces of $V$. It is an important tool in the study of $[1,c]_T$. When $W$ is infinite, the moved space of an element $w\in [1,c]_T$ no longer has dimension $\ell_T(w)$ in general, and distinct elements may have the same moved space. We propose a replacement for the moved space of an element $w\in [1,c]_T$ in an arbitrary Coxeter group, that we call absolute moved space of $w$. This subspace $\mathsf{AM}(w)$ of $V$ always contains $\mathsf{Mov}(w)$ and has dimension equal to $\ell_T(w)$, and distinct elements have distinct absolute moved spaces. This allows us to derive several properties of noncrossing partition posets that hold in full generality, and to show that the natural map from $[1,c]_T$ to reflection subgroups of $W$, which to $w\in [1,c]_T$ associates the subgroup $P(w)$ generated by reflections lying below $w$ in the absolute order, is always injective. Among others, we also derive a new proof of the lattice property of $[1,c]_T$ when $W$ has rank three, and exhibit infinitely many new examples of infinite Coxeter groups of rank four and choices of Coxeter elements for which $[1,c]_T$ fails to be a lattice.

We show that the Knörrer functor induces an equivalence ${\underline{\operatorname{MCM}}}(R) \simeq {\underline{\operatorname{MCM}}}(A)$ of stable categories of maximal Cohen-Macaulay modules, where $R = S/(f)$ is a complete hypersurface ring and $A = S[\![u,v]\!]/(f+uv)$ is the hyperbolic extension of $R$. No hypothesis is placed on the residue field or on its characteristic, $f$ need not define an isolated singularity, and $S$ need not contain a field. This is in contrast to the iterated double branched cover $R^{\sharp\sharp} = S[\![z,w]\!]/(f+z^2+w^2)$, which is stably equivalent to $R$ only in characteristic not equal to $2$. We also give a direct construction of a free resolution identifying $\operatorname{syz}_2^A(M)$ with the image of $M \oplus \operatorname{syz}_1^R M$ under the functor, and an example in characteristic two in which the corresponding statement for the double branched cover $S[\![z]\!]/(f+z^2)$ fails.

Let $K$ be an algebraically closed field of odd characteristic $p$, let $G= {\rm SL}_2(K)$, and let $V$ be the natural representation of $G$. Let $b_k$ denote the number of $G$-indecomposable factors of $V^{\otimes k}$, counted with multiplicity, and let $\delta_p=1-\log_{p^2}\!\bigl(\tfrac{p+1}{2}\bigr)$. Then there exists a smooth, strictly positive, multiplicatively $p^2$-periodic function $\omega(t)$ such that $b_k$ is asymptotic to $\omega(k)k^{-\delta_p}2^k$. We also show that $t^{-\delta}\omega(t)$ arises as the limiting density of renormalized convolutions of rescaled copies of a positive weight $3/2$ theta function, obtained from the boundary heat flux of the Dirichlet heat kernel on $(0,p)$.

We construct a correspondence between ordinary irreducible characters of height zero above the generalized Dade-Glauberman-Nagao correspondence for blocks with central defect. The correspondence is equivariant under group and Galois automorphisms and satisfies a block relation between $\mathcal{H}$-triples. For each pair of corresponding characters, one pair of associated projective representations realizes matching factor sets, equal central scalars, synchronized mixed comparison functions and compatible block correspondences for every intermediate group. The construction uses the multiplicity algebra over the group algebra of the central defect subgroup, a normalized integral lift of a residual semilinear realization, and a graded corner isomorphism. It treats all central-character fibers, including those with nontrivial central character. We also obtain a form suited to normal $p$-sections in an Alperin-McKay-Navarro reduction.

The construction of absolutely irreducible $KG$-modules for a field $K$ and a finite group $G$ is a basic problem in representation theory. For the symmetric group $\operatorname{S}_n$, Gordon James gave an explicit solution using subgroup data associated to a $\mu$-tableau $t$ for partitions $\mu\vdash n$. Here, we want to investigate in which way James's method can be extended beyond symmetric groups. We identify suitable subgroup data from a finite group and use it to construct a module such that an associated factor module is absolutely irreducible or zero.

We introduce and study the algebraic Dirac operator for the Cartan motion group associated with a real reductive Lie group. We establish its fundamental properties, including a square formula involving a natural Casimir-type element, define the corresponding notion of Dirac cohomology, and prove a strong form of Vogan's conjecture: a module in the Dirac series is completely determined by its Dirac cohomology. We explicitly determine the Dirac series of a Cartan motion group. We explain how the Dirac theory of a Cartan motion group can be obtained as the zero fiber of the recently introduced Dirac theory for the algebraic deformation family of the corresponding real reductive group.

Let $\overline{\mathfrak{F}}$ be the group class obtained from the class $\mathfrak{F}$ of finite groups by iterated applications of Kropholler's operation ${\scriptstyle{{\bf LH}}}$ and Talelli's operation $\Phi$. We describe the structure of the cokenels of acyclic complexes of projective modules over the group algebra of $\overline{\mathfrak{F}}$-groups with coefficients in a commutative ring. If $G$ is an $\overline{\mathfrak{F}}$-group, we show that any acyclic complex of projective (respectively, flat) $\mathbb{Q}G$-modules is contractible (respectively, pure acyclic). If $G$ is torsion-free, the same conclusions hold for acyclic complexes of projective or flat $\mathbb{Z}G$-modules. Analogous results are valid for acyclic complexes of injective modules. We present some applications regarding modules that admit complete resolutions, groups with periodic cohomology after some steps and the relation between Gorenstein and ordinary homological dimensions for modules over group algebras.

We construct relative suspension for the exact category generated by a support $\tau$-tilting module and characterize higher delooping by the units of the resulting adjunction, which are the generalizations of Gélinas's results in Adv. Math.(2022). We prove that comparison with ordinary higher delooping requires a shift of one in the index. More precisely, for every $k,d\ge1$, there is a module whose ordinary $k$-delooping level is zero and whose relative level is $d$. We also prove that tensor induction preserves the relative constructions and higher delooping levels on induced modules. As an application, we study cyclic Nakayama algebras with arbitrary local coefficients and automorphism twists. Their left and right little and big finitistic dimensions and ordinary and derived delooping levels all equal the finitistic dimension of the underlying Nakayama algebra.

In this article, we introduce a new rewriting method for computing hom-bases of diagrammatic categories. Our approach is based on stratified presentations, obtained by partitioning an equational presentation into strata according to the rewriting behavior of its relations. We establish modularity principles for termination and confluence in stratified rewriting systems with coefficients in a ring. We then introduce a completion procedure that transforms a stratified presentation into a normalized one, in which mixed interactions between successive strata are absorbed into the normalized rewriting rules, thereby inducing modularity of confluence. This leads to a stratified hom-basis theorem for diagrammatic categories. We illustrate the method by applying it to the $q$-Schur category, for which we construct a new hom-basis.

Let \(X\) be a rationally smooth complex affine variety with a torus action and an attractive fixed point \(x\). Suppose that \(X\setminus\{x\}\) is \(p\)-smooth and that its integral equivariant cohomology has no \(p\)-torsion. We prove that the order of its total \(p\)-primary cohomology torsion is the \(p\)-part of the reduced numerator of the equivariant multiplicity at \(x\). In particular, this resolves the torsion-order conjecture of Juteau–Williamson and its local version for normal slices in Schubert varieties, using the equivariant torsion-freeness result of Fiebig–Williamson.

Let $G$ be a reductive group over a number field $F$. We define $C^*_{\rm aut}(G(\mathbb A))$, the automorphic $C^*$-algebra of $G$, to be the $C^*$-algebraic image of the full automorphic representation of $G(\mathbb A)$ on $L^2(G(F)\backslash G(\mathbb A))$. Using the Langlands spectral decomposition with respect to discrete Levi data, we construct an injective $*$-homomorphism \[ C^*_{\rm aut}(G(\mathbb A)) \longrightarrow \bigoplus_{[M,\sigma]} K_{C_0(\widehat{A_M})} \left( \operatorname{Ind}_P^G C_0(\widehat{A_M},H_{M,\sigma}) \right)^{W(G,M,\sigma)}, \] where $[M,\sigma]$ ranges over the associate classes of discrete Levi data. When $G$ is $\operatorname{GL}(n)$ or an inner form of it, we prove that this map is an isomorphism of $C^*$-algebras.

Let $\mathcal{O}$ be a complete discrete valuation ring with a finite residue field. We introduce a simple framework of constructing smooth representations of $\mathrm{GL}_n(\mathcal{O})$ modulo the knowledge of nilpotent orbits, called Jordan induction, which enjoys several remarkable properties: It yields only irreducible representations, it yields all the even level irreducible representations, and up to conjugation it yields non-isomorphic representations. In the even level case, this gives an explicit realisation of Hill's analogue of Lusztig's Jordan decomposition, as well as a vast generalisation of Gérardin's construction for $\mathrm{GL}_n(\mathcal{O})$. As a simple application, we construct an explicit section to the orbit map. We also propose a conjecture linking Jordan induction and Lusztig induction, aiming at generalising the algebraisation theorem obtained in recent joint works with Stasinski.

We prove Keller's conjecture for every split finite-dimensional algebra over an arbitrary field.

We give a computer-assisted proof of Frenkel's root-multiplicity bound for the rank-three hyperbolic Kac–Moody Lie algebra $A_1^{++}$. For every root $\alpha$, we prove $\dim g_\alpha\le p(1-(\alpha,\alpha)/2)$, where $p$ is the ordinary partition function. The proof combines exact affine characters with a coefficientwise logarithmic majorant derived from parabolic homology. After Weyl reduction, analytic estimates establish the inequality on infinite regions of large depth or large level. The remaining finite region is verified by computer. Together with the known low-level formulas, the assembly of these estimates establish the conjecture for every root of $A_1^{++}$.

For a finite group $G$, define the field $\mathbb{Q}(G) := \mathbb{Q}(\{\chi(g) : \chi \in \operatorname{Irr}(G) , g \in G\})$. In this article we compute generating sets for the fields $\mathbb{Q}(G)$ when $G$ is a finite general or special linear group, a finite general or special unitary group, or one of the simple groups $\operatorname{PSL}_n(q)$ or $\operatorname{PSU}_n(q)$. We then apply these results to classify all number fields $F$ of degree $2$ or $3$ over $\mathbb{Q}$ such that $F = \mathbb{Q}(S)$ for some non-abelian simple group $S$.

Let $k$ be a finite field. We consider the category of all functors $\mathcal{F}_n$ from the category of $k$-vector spaces of dimension $\leq n$ into the category of all $k$-vector spaces. We compare its derived category to the derived category of representations of the general linear group $GL_n(k)$ using the extension by zero functor procedure. We upgrade the well-known recollement of abelian categories into the recollement of the corresponding derived categories. Then we obtain some translations of the cohomology the general linear group over finite field. We also compare $\mathcal{F}_n$ with the category of polynomial functors $\mathcal{P}_{d,n}$, giving further reformulations over big finite fields.

In this article, we study derived Hecke algebras with coefficients in torsion-free profinite rings regarded as discrete rings. We compare each such algebra with the degreewise inverse limit of the derived Hecke algebras with coefficients in finite rings and with its subalgebra consisting of elements with uniformly finite double-coset support. Under cohomological comparison hypotheses, the natural map to this subalgebra has image given by its degree-zero part and its positive-degree torsion. We describe the kernel in terms of continuous cohomology and prove that it is a divisible square-zero ideal annihilated by every positive-degree element. These comparisons are compatible with Yoneda products, and the action on torsion Ext classes factors through the image. We compute the resulting algebras for the additive group $\mathbb{Q}_p$ and determine the underlying graded abelian groups for $(\operatorname{GL}_2(\mathbb{Q}_p),\operatorname{GL}_2(\mathbb{Z}_p))$ with coefficients in $\mathbb{Z}_p$, for $p>3$. The examples exhibit distinct effects of the coefficient topology and the support condition.