Papers

It was conjectured that no presilting objects exist in the singularity category of a finite-dimensional algebra over a field except the zero object. In this note, we give counterexamples to this conjecture for algebras over fields of positive characteristic.

Strongly bounded objects in a weakly approximable triangulated category are known to relate closely to global dimension and to play a significant role in the uniqueness problem for triangulated enhancements. In this paper, we investigate when the full subcategory of strongly bounded objects admits a bounded $t$-structure. Our main result, under a finiteness condition termed the finite strong finitistic dimension, states that this happens exactly when the subcategory agrees with the full subcategory of bounded objects in the ambient triangulated category. In that case, the bounded $t$-structure is unique up to equivalence; even more, the uniqueness holds unconditionally on the full subcategory of bounded objects.

For a vertex operator algebra $V$ and a certain category of its modules, we propose a construction for spaces of conformal blocks organized into an open-closed modular functor with singularities. This is inspired by the idea of implementing directly from the start the principle of holomorphic factorization. More precisely, using the strategy of modular extension introduced by Costello and developed further in our previous work, we build for each surface $\Sigma$ with at least one boundary component per path component and specified boundary labels attached to marked intervals or boundary circles a representation $\Omega_V(\Sigma;-)$ of the mapping class group of $\Sigma$. The construction can be described explicitly on generating Dehn twists. This approach is a priori independent from other constructions based on algebraic geometry or topological techniques involving e.g. surgery, but we include an overview over the available comparisons. In the special case in which the module category of $V$ is a not necessarily semisimple modular category $\mathcal{A}$, the spaces $\Omega_V(\Sigma)$ are equivalent to the string-net spaces for $\mathcal{A}$ and hence to the modular functor for the Drinfeld center $Z(\mathcal{A})\simeq \bar{\mathcal{A}}\boxtimes\mathcal{A}$. However, the construction of $\Omega_V$ in this paper has the advantage of being available beyond rationality, rigidity, self-contragredience and finiteness. Moreover, we prove that $\Omega_V$ satisfies excision, is finite-dimensional in the $C_2$-cofinite case and produces in genus one a generalization of the elliptic double of Brochier-Jordan. We prove for the triplet $\mathcal{W}_{2,3}$ with non-exact fusion product that the boundary conditions introduced by Gaberdiel-Runkel-Wood produce, as expected by these authors, correlation functions, provided that one uses the notion of a modular functor with singularities that we develop.

Quantum Fourier transforms (QFTs) are essential primitives in quantum algorithms. While abelian groups admit efficient QFT circuits, with circuit size polynomial in the logarithm of the group order, efficient constructions are known for relatively few non-abelian families. We develop two new approaches to QFT circuit construction, based on Mackey theory and Clifford theory, respectively, and use them to show exponential improvement in circuit cost for specific group families. Using the Mackey-theoretic approach, we obtain explicit quantum circuits for the QFT over $\mathrm{GL}_2(F_q)$ that scale polynomially in $\log q$, rather than polynomially in $q$. Using the Clifford-theoretic approach, we obtain QFT circuits for wreath products $F\wr S_n$, whose cost depends on the cost of a QFT over $F$ and the size of its representation registers. This removes the restriction $|F|=\operatorname{poly}(n)$ required by previous generic constructions and can yield exponential improvements when $F$ itself has an efficient QFT. Together, these methods provide new systematic tools to construct QFTs for broad classes of finite groups.

Quantum Fourier transforms (QFT) for general groups were recognized to be fundamental already early in the field. A canonical example of non-abelian QFT for the symmetric group was outlined by Beals (1997). Later, a more detailed analysis of this algorithm was carried out by Kawano and Sekigawa (2016). In this paper, we revisit that construction. After a careful analysis, we revise their gate complexity to $\widetilde{\mathcal{O}}(n^{3.5})$ and circuit depth to $\widetilde{\mathcal{O}}(n^3)$. Moreover, we observe that their construction is not optimal in the choice of transversal elements, so we propose simpler realization of the symmetric group QFT.

We prove Mühlherr's Twist Conjecture: any two angle-compatible Coxeter generating sets of a Coxeter group differ by a finite sequence of elementary twists and a conjugation. Combined with earlier work of Howlett-Mühlherr and Marquis-Mühlherr, this completes the resolution of the Isomorphism Problem for Coxeter groups. A further consequence is that ${\rm Aut}(W)$ is finitely generated for every Coxeter group $W$, and there is an algorithm producing a finite set of generators for ${\rm Aut}(W)$ starting from any Coxeter matrix. Of the vast literature on the Twist Conjecture, we utilise only two results in an essential way: strong rigidity of $2$–spherical Coxeter systems, due to Caprace and Mühlherr, and the framework of markings and hierarchies developed by Caprace and Przytycki for the twist-rigid case. We also exploit in a fundamental way some soft ideas from JSJ theory and an observation of Mihalik-Tschantz on splittings of Coxeter groups. No form of AI was used in the writing of this manuscript, nor in the research that it presents.

In this short note, making use of the theory of noncommutative motives, we prove that the Grothendieck classes of the twisted symplectic flag varieties are linearly independent and moreover homogeneous-quadratic independent in the Grothendieck ring of algebraic varieties.

Lusztig-Vogan categories are categorifications of the principal block of the Lusztig-Vogan module over the Hecke algebra, which captures information about characters of irreducible admissible representations of a real reductive group. Lusztig-Vogan categories can be constructed as module categories over Soergel bimodules. In this paper, we describe the structure of the rank 2 Lusztig-Vogan categories corresponding to equal rank real groups. More precisely, we classify indecomposable objects and describe the action of generating Soergel bimodules, recovering the $W$-graph of the underlying Lusztig-Vogan module. We also provide an algorithm which completes this procedure for arbitrary finite rank Lusztig-Vogan categories, including those which do not correspond to a real reductive group.

We give a uniform realization of Kirillov-Reshetikhin crystals $B^{r,s}$, in terms of PBW crystals, for all affine types, $s \geq 1$, and nodes $r$ in the orbit of the $0$ node in the Dynkin diagram. Our proof is almost uniform except we prove some technical lemmas by first proving the case when the $r$-th fundamental weight $\Lambda_r$ is minuscule by using properties of minuscule crystal and then extend this to the case when $\Lambda_r$ is cominuscule using virtual crystals.

Let $A$ be an Artin algebra and let $T$ be a self-orthogonal $\tau$-tilting right $A$-module. Set $I=\Ann_A(T)$. We prove that \[ \Hom_A(I,T)=0=\Ext_A^n(I,T)\qquad(n\geq 1). \] It follows that the torsion class generated by $I$ is Hom-orthogonal to $\Sub T$. This separation yields faithfulness criteria expressed through ideal tops, socles, and projective supports. We also prove a finitistic-dimension obstruction: if $B=\End_A(T)^{\rm op}$ has finite little finitistic dimension, then $\Fac T\cap{}^{\perp_{\geq0}}T=\{0\}$. As an application, the self-orthogonal $\tau$-tilting conjecture holds for radical square zero Artin algebras. Finally, the support criteria apply to semisimple-source triangular matrix algebras with arbitrary local terminal blocks.

We study deep loci for the reduced-word Berenstein-Fomin-Zelevinsky cluster atlas on type-A double Bruhat cells. As defined in CGSS24, for a seed collection on a cluster variety X, the deep locus is the complement of the union of all corresponding cluster tori. We focus on double Bruhat cells G(u,v) for G = SL(n). Our main results are several theorems describing the structure of these deep loci, including their interaction with the Poisson structure, the relationships between the deep loci of different cells, and a tracking formula that allows us to compute the reduced-word deep locus of a cell from a lower-dimensional one. The paper also includes explicit low-rank calculations for double Bruhat cells in SL(3) and Borel double Bruhat cells in SL(4). Finally, we discuss the relationship between these results and some general properties of deep loci.

Let $\mathcal O_{2A_1}$ be the next-to-minimal nilpotent orbit of the complex simple Lie algebra of type $E_7$. Let $\mathfrak p_7$ be the maximal parabolic subalgebra corresponding to the simple root $\alpha_7$, and let $\mathfrak u^+$ be its nilpotent radical. We prove that the affinization $T^*(\mathcal O_{2A_1}\cap\mathfrak u^+)^{\mathrm{aff}}$ of the cotangent bundle of $\mathcal O_{2A_1}\cap\mathfrak u^+$ is isomorphic to the nilpotent orbit closure $\overline{\mathcal O}_{2A_1}$ as affine varieties.

We study the singularity categories of simple singularities of the same dimension over an algebraically closed field of positive characteristic, and show that, as in characteristic zero, these categories are not equivalent as triangulated categories unless the underlying singularities are analytically isomorphic. In contrast to the characteristic zero case, simple singularities in positive characteristic cannot be distinguished solely from the Auslander-Reiten quivers of their singularity categories. To address this, we extend to positive characteristics a theorem by Hua and Keller, which asserts that the 0th Hochschild cohomology of the dg singularity category of an isolated hypersurface singularity in characteristic zero is isomorphic to the Tyurina algebra of the defining polynomial. Furthermore as an application, we determine the condition for the singularity category of a rational double point (i.e., a simple singularity of dimension two) to be standard. We prove that such a category is standard if and only if the defining polynomial is weighted homogeneous.

The rows and columns of the character table of the symmetric group $S_n$ are both naturally indexed by partitions of $n$. Let $D(n)$ denote the number of conjugacy classes of $S_n$ whose column contains no zero entry. The identity column is always zero-free, so $D(n)\geq 1$. It is known that $D(n)\ll n^2$. We prove that $D(n)\ll n^{3/4}$. Second, we prove for almost all positive integers $n$ that $D(n)\ll_B n^{1/2}(\log n)^B$ for every $B>5/6$, with a quantitative bound for the exceptional set, using work of Matomäki and Radziwill. Finally, we offer a heuristic supporting our conjecture that $D(n)\ll_{\varepsilon} n^{\varepsilon}$. AxiomProver formalized the results in this paper in Lean assuming preexisting literature.

This paper concerns homological algebra based on $Q$-shaped diagrams, where $Q$ belongs to a certain class of small categories. We will show that the Gorenstein property of noetherian commutative rings is characterised by the condition that acyclicity coincides with total acyclicity for $Q$-shaped diagrams of suitable classes of modules. Many special cases occur as corollaries, for instance $N$-complexes. This generalises a classic result by Iyengar and Krause to the programme of $Q$-shaped derived categories, which builds on an insight of Iyama and Minamoto. We prove our result using an adjoint triple of functors relating $Q$-shaped diagrams to chain complexes. One of the functors was introduced by Jasso, and the whole triple generalises the compression, cocompression, and expansion functors for differential modules introduced by Avramov, Buchweitz, and Iyengar and by Nkansah. We consider the adjoint triple to be the main contribution of this paper.

Let $\pi$ be a set of primes and let $G$ be a finite $\pi$-separable group. We prove that the Cartan matrix $C$ for the $\pi$-partial characters of $G$ is equivalent over the integers to a matrix $\operatorname{diag}(|\mathbf{C}_G(x)|_{\pi'})$, where $x$ runs over a set of representatives of the $\pi$-classes of $G$. In particular, we prove that $\det C = \prod |\mathbf{C}_G(x)|_{\pi'}.$

Let $U$ be a Sylow $p$-subgroup in a classical orthogonal group over a finite field with $q$ elements of characteristic $p$ large enough. The coadjoint orbits of the group $U$ play the key role in the description of irreducible complex characters of $U$. In the paper, we provide a classification of such orbits of pre-maximal dimension. As a corollary, we compute the number of all orbits mentioned above. It turned out that each of these numbers is a polynomial in $q - 1$ with integer non-negative coefficients, which agrees with the Isaacs' conjecture.

Let $U$ be a Sylow $p$-subgroup in a classical group over a finite field with $q$ elements of characteristic $p$ large enough. The coadjoint orbits of the group $U$ play the key role in the description of irreducible complex characters of $U$. In the paper, we provide a classification of such orbits of pre-maximal dimension for symplectic groups and orbits of maximal dimension for orthogonal groups. As a corollary, we compute the number of all orbits mentioned above. It turned out that each of these numbers is a polynomial in $q - 1$ with integer non-negative coefficients, which agrees with the Isaacs' conjecture.

A web is a plane graph encoding an invariant vector in a tensor product of fundamental representations of a quantum group. A stranding of an $\mathfrak{sl}_n$ web is a system of colored oriented curves recording one monomial of the vector it encodes. This article focuses on identifying and constructing leading term strandings, those recording the leading term of a web's vector with respect to a lexicographic order on monomials. We show that every open strand of a leading term stranding is clockwise, so the boundary data of such a stranding is recorded by a row-strict tableau. From a row-strict tableau, we construct a web with a prescribed leading term, and the resulting webs form a web basis. This non recursive construction recovers Fontaine's bases and produces others. For $\mathfrak{sl}_3$ webs with no flat vertices, we identify a leading term stranding using the depths of the faces of the web. Finally, we ask whether any two strandings of a web are connected by a sequence of local moves called strand reversals. We answer this question affirmatively for $\mathfrak{sl}_3$, extending a known result for closed webs to webs with boundary; in particular, a leading term stranding of any $\mathfrak{sl}_3$ web is always reachable from an arbitrary stranding.

In this article, we construct noncommutative cluster varieties, $\mathcal{A}_{R,S}$, for each reduced root system $R$ and marked surface $S$ simultaneously generalizing the cluster varieties of Fock-Goncharov, Li, Goncharov-Shen, Berenstein-Retakh, Goncharov-Kontsevich, and our previously introduced polygonal cluster algebras. Additionally, we define a large class of algebraic groups, we call Jordan split groups. Given a reduced root system $R$ and a family of Jordan algebras, the Lie algebra for $G$ is constructed by unifying the Tits-Kantor-Koecher construction for a single Jordan algebra with the construction of a split Lie algebra. The notion of Jordan split groups is closely related to a grading of its Lie algebra by the root system $R$. We show that these gradings are usually induced by a choice of standard parabolic subalgebra $\mathfrak{p}_\Theta$ and we classify $R$-graded pairs $(G,\Theta)$ via a condition depending only on the subset $\Theta\subset \Delta$ of the set of simple roots. Next, we define Jordan algebra points of $\mathcal{A}_{R,S}$ which parameterize $G$-local systems on $S$ with boundary decoration related to cosets $G/U_\Theta$ when $G$ is Jordan split of type $R$. When $S$ is a disk, points of $\mathcal{A}_{R,S}$ parameterize configurations of decorated flags. We use this to give noncommutative cluster structures on the double $R$-Bruhat cells of $G$, generalizing the cluster algebras of Berenstein-Fomin-Zelevinsky. When each Jordan algebra is formally real, we say that $G$ has a positive structure with respect to $\Theta$. This defines a positive semigroup in $G$. For real algebraic groups, the pairs $(G,\Theta)$ which have positive structures are exactly those which admit a positive structure as defined by Guichard-Wienhard and we give algebraic proofs of many of the properties of positive configurations of flags and of positive representations.

We present a simple combinatorial method for computing the socle of representations of $\mathrm{GL}_n(F)$, where $F$ is a local non-Archimedean field, parabolically induced from two ladder representations satisfying a permissibility condition. We adapt recent approaches to such problems using the classical Robinson–Schensted–Knuth correspondence (RSK) and introduce a new variant of RSK that we call mixed-RSK. Some combinatorial properties of this map are investigated and then used to resolve a conjecture of Erez Lapid.

We investigate the form of the Saito metric on quotients of the reflection representation of an extended affine Weyl group, restricted to an arbitrary stratum of its discriminant. We compute the Saito determinant for all strata and Dynkin types and show that it is divisible by a product of q-analogues of restricted roots for the associated hyperplane arrangement. Our results simultaneously generalise the factorisation in the Weyl denominator formula to higher codimension strata, and provide q-analogue refinements of results of Antoniou-Feigin-Strachan for Coxeter groups.

In this paper, we construct a lifting operator from the Grothendieck group of admissible Harish-Chandra modules of $G =\mathrm{SO}_{2n}(\mathbb C)$ (resp. $\mathrm{Sp}_{2n}(\mathbb C)$) to that of genuine representations of $\mathrm{Spin}_{2n}(\mathbb C)$ (resp. $\mathrm{Spin}_{2n+1}(\mathbb C)$). We determine the lift of the sum of special unipotent representations attached to any ${}^{\vee}\mathcal O \subseteq {}^{\vee}\mathfrak g$ explicitly. In particular, the representations occurring in these lifts, if nonzero, are genuine unipotent representations of complex Spin groups and are unitary. As a consequence, the lifting operator preserves unitarity on a large class of unitary representations.

Under the assumption that the irreducible unitary representation $\pi$ is infinite dimensional, this paper improves the Helgason-Johnson bound in 1969 for $Sp(p, q)$.

Let $\mathcal C$ be an $(n+2)$-angulated category. Zhou proved that, when $n$ is odd, if the Auslander-Reiten $(n+2)$-angles generate the relations for the Grothendieck group of $\mathcal C$, then $\mathcal C$ is locally finite. Whether the corresponding statement remains valid for even $n$ is still open. In this paper, we give a partial affirmative answer to this problem by establishing a sufficient condition under which the same implication holds for even $n$. We further show that our sufficient condition is satisfied by a broad class of examples, thereby demonstrating that the result extends well beyond isolated cases.

We prove a Hall $\sigma$-subgroup analogue of the McKay conjecture for $\pi$-separable groups proposed by G. Navarro. This result simultaneously generalizes the classical McKay conjecture and its $\pi$-separable version. More precisely, if $G$ is $\pi$-separable, $p$ is a prime and $\sigma=\pi\cup\{p\}$, then it is known that there exists $H$ a Hall $\sigma$-subgroup of $G$. In this case, we prove that $|{\rm Irr}_{\sigma'}(G)|=|{\rm Irr}_{\sigma'}(\mathbf{N}_{G}(H))|$.

We show that the nilpotent cone associated to a simply connected semisimple algebraic group, together with all its crepant projective partial resolutions, can be constructed as Hamiltonian reductions of the affine closure of the cotangent bundle of base affine space for suitable choices of stability parameter of a maximal torus. We also realize the base change of the universal Poisson deformation of each of these crepant projective partial resolutions as the GIT quotient of the affine closure of the cotangent bundle of base affine space at the same stability parameter.

We present a framework for studying the moduli space of linear determinantal representations for flat families of complex projective cubic surfaces across singularity boundaries. First, we deal with the classical deformation theory where a surface S_0 possesses a rational double point (RDP) of type E_6. By establishing a global simultaneous resolution over a finite ramified Galois covering of the global parameter slice, we realize the relative moduli space H_N as a diagonal Weyl quotient (h x R)/W(E_6). Second, we extend this classification to the critical boundary configuration where S0 contains a unique isolated simple elliptic singularity of type E~_6. Because the local monodromy group becomes infinite, the classical simultaneous resolution framework completely breaks down. We bypass this obstruction by constructing a geometric substitute Z derived from the filtration of Looijenga's invariant algebra of affine Weyl groups. We also show that the vector bundle Z decomposes into a direct sum of line bundles over the structural elliptic curve E, with degrees explicitly matching the negative Coxeter marks of the highest root of E_6. Utilizing Riemann's extension theorem and vector bundle rigidity over elliptic curves, we settle our isomorphism result, proving that the moduli space H-bar representing the semiuniversal family is globally isomorphic to the total space of Z.

We show that a contravariantly finite resolving subcategory over a henselian local ring must be the subcategory of free modules or the whole module category or the subcategory of maximal Cohen-Macaulay modules. This removes the Gorenstein assumption from a theorem of Takahashi. We also show that a resolving subcategory of finite type other than the subcategory of free modules must be the subcategory of maximal Cohen-Macaulay modules. As a consequence, every Cohen-Macaulay local ring of finite CM type is uniformly dominant, thereby establishing a stronger form of a conjecture of Takahashi.

We classify all characters of quantum symmetric pair coideal subalgebras for generalised Satake diagrams of finite and affine type.