Papers

Bastidas et al. (2025) highlighted the importance of exploiting group symmetries to reduce computational overhead and unify simulation methods across both condensed matter physics and ab initio quantum chemistry. I discuss the discrete character transform, which is the group Fourier transform restricted to $Z(\mathbb{C}[G])$, the center of the group algebra, where class functions form a commutative convolution algebra and irreducible characters are its spectral basis. The normalized character table of a non-abelian group can be obtained through a restriction operator, which means an efficient quantum character transform for any finite group with an efficient quantum Fourier transform. I give complementary quantum algorithms for class convolution based on direct class-sum block encoding and spectral implementation in the character basis, explaining their applications to symmetry-adapted many-body dynamics, molecular and spectroscopic symmetry analysis, and quantum walks, as well as characterizing the central information used in hidden-subgroup problems, including its limitations for graph isomorphism.

We develop a support theory for wide subcategories of essentially small Krull–Schmidt \(n\)-abelian categories. We construct spectra associated with families of wide subcategories and establish the corresponding support and radical theories. In particular, we characterize prime wide subcategories as the completely meet-irreducible elements of the lattice of wide subcategories and prove that every wide subcategory is recovered from the prime wide subcategories containing it. We further show that any classifying support datum is recovered, together with its object supports, from the canonical prime spectrum. Finally, we provide a \(2\)-abelian example admitting no classifying support datum.

Assuming a smoothness property for mixed-characteristic shtuka spaces and hyperbolic localization for twisted sheaves, we determine the inertial restriction of the Fargues-Scholze $L$-parameter for irreducible depth-zero representations. Let $E$ be a non-archimedean local field of residue characteristic $p$, let $G$ be an unramified connected reductive group over $E$, and let $\pi$ be an irreducible smooth depth-zero representation of $G(E)$ over $\overline{\mathbb{F}}_{\ell}$ where $\ell$ is a prime number which does not divide $p|\pi_0Z(G)|$. Let $\varphi_{\pi}^{\operatorname{FS}}$ be the Fargues-Scholze $L$-parameter associated to $\pi$. We describe the restriction of $\varphi_{\pi}^{\operatorname{FS}}$ to the inertia subgroup in terms of Deligne-Lusztig theory. We prove this by computing the inertia action on the cohomology of shtuka spaces with depth-zero level structure. As an application, we prove that the Fargues–Scholze $L$-parameters agree with DeBacker–Reeder $L$-parameters over $\overline{\mathbb{Q}}_{\ell}$.

In this paper, we extend the notion of a removable subcategory to $n$-exact categories, and prove that an ideal quotient by a subcategory of projective-injectives inherits an $n$-exact structure from the original category if and only if the subcategory is removable. In the Frobenius case, the quotient remains Frobenius, and the stable $(n+2)$-angulated categories are equivalent. We give a sufficient condition for a subcategory of projective-injectives to be removable. We also show that the ideal quotient construction is compatible with cluster tilting. Finally, we apply the construction to $2$-cluster tilting subcategories of the category of ACM bundles on a Geigle–Lenzing projective plane, obtaining quotients for both natural and relative $2$-exact structures.

This paper studies the closures of the double Bruhat cells of symmetric spacess in its equivariant embeddings. We call them large double Bruhat varieties. Our aim is to establish the double Bruhat stratification as a natural setting for extending the geometry of Richardson varieties to symmetric spaces. For the wonderful compactification of the associated adjoint symmetric space, we show their scheme-theoretic intersections with the unique closed orbit are reduced unions of projected Richardson varieties. This connection with Richardson geometry on partial flag varieties supports large double Bruhat varieties as a natural class of subvarieties in symmetric-space embeddings. In positive characteristic, we prove that every equivariant embedding admits a Frobenius splitting compatible with all large double Bruhat varieties and all boundary divisors. These varieties are strongly F-regular, and globally F-regular when the ambient embedding is projective. In characteristic zero, they are of strongly F-regular type, and of globally F-regular type in the projective case; in particular, they are normal, Cohen-Macaulay, and have rational singularities.

Let $G$ be a split real semisimple group of adjoint type. We prove that the cellular decomposition of the totally nonnegative part of its wonderful compactification is a regular CW complex, confirming a conjecture of Galashin, Karp, and Lam. In particular, the closure of every cell is homeomorphic to a closed ball. The proof combines a Bruhat atlas compatible with total positivity and a CL-labeling of the augmented face poset. The atlas models the strata locally with open Richardson varieties in the flag variety of a larger Kac-Moody group and identifies their totally nonnegative cells.

We determine the representation type and Schurian finiteness of the Tits algebra of a finite central real hyperplane arrangement over an algebraically closed field. The algebra is representation-finite when the arrangement has at most two hyperplanes, tame when it has three, and wild when it has at least four. Moreover, it is Schurian-finite if and only if the arrangement has at most two hyperplanes. Both classifications hold over algebraically closed fields of arbitrary characteristic. The proof combines Saliola's bound-quiver presentation with explicit module constructions, convex subquivers, essentialization, and deletion. In the $\mathbb{R}^2$ case, we describe all eleven indecomposable modules for two lines and construct a full exact embedding of modules over a free algebra for at least four lines.

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with root system $\Phi$. A subset $S\subseteq\Phi$ is closed if $\alpha,\beta\in S$ and $\alpha+\beta\in\Phi$ imply $\alpha+\beta\in S$. For finite and affine Kac-Moody algebras, this closure property is known to extend from pairs to arbitrary finite sums that are roots. We prove that every closed subset of the root system of any symmetrizable Kac-Moody algebra is closed under arbitrary finite sums that are roots. The proof uses real-root strings and simultaneous separation of positive imaginary roots, together with an addition lemma for fundamental imaginary roots. As an application, we extend a necessary condition for wideness of regular subalgebras from finite and affine types to all symmetrizable Kac-Moody algebras: the symmetric closure of the roots associated to a wide regular subalgebra is the entire root system.

Let $F/\mathbf Q_p$ and $E/\mathbf Q_p$ be finite extensions, and let $G=\mathrm{GL}_2(F)$ or $D_F^\times$, where $D_F$ is the quaternion division algebra over $F$. Let $\Pi$ be an admissible Banach $E$-representation of $G$ whose locally $\mathbf Q_p$-analytic vectors $\Pi^{\mathrm{an}}$ have an infinitesimal character. If $p>2$ or $F=\mathbf Q_p$, we prove that $\Pi^{\mathrm{an}}$ and $\Pi$ have finite length in their respective admissible categories.

Let $\mathcal O$ be the ring of integers of a non-Archimedean local field with uniformizer $\varpi$, and let $\mathcal O_\ell=\mathcal O/\varpi^\ell\mathcal O$ for $\ell\geq1$. In this article, we study representations of $\mathrm{GL}_n(\mathcal O_\ell)$ parabolically induced from characters of the Borel subgroup $\mathrm{B}_n(\mathcal O_\ell)$. For $n=3$, we establish a criterion for the irreducibility of these induced representations in terms of the relative conductors of the inducing characters. Motivated by this result, we formulate a conjecture characterizing the irreducibility of such representations for arbitrary $n$. We further show that when all inducing characters are equal, the resulting induced representation is not multiplicity-free, implying that the corresponding triple $(\mathrm{GL}_n(\mathcal O_\ell),\mathrm{B}_n(\mathcal O_\ell),\chi)$ is not a Gelfand pair. Finally, we study the irreducible decomposition of these representations for $\mathrm{GL}_3(\mathcal O_\ell)$ when two relative conductors attain their maximal values. Using the decomposition theory for $\mathrm{GL}_2(\mathcal O_\ell)$, we describe the irreducible constituents and determine their dimensions.

We generalize Nakajima's theory of Hilbert schemes of points on surfaces to Hilbert schemes of points on orbisurfaces, and obtain initial factorization based on stratification by stabilizers. While the usual formula is retained for the trivial stabilizer stratum and for those strata of codimension $1$, genuine difficulties arise with stacky points strata. In the local picture of an isolated stacky point, the theory reduce to the study of $\Quot^\bfv[\CC^2/G]$ for a small finite subgroup of $GL_2(\CC)$. The generating series for their total cohomology is a geometric correction of the generating series for total cohomology of ordinary smooth surface schemes. In addition, we obtain an action of the Borel-Moore homology of the punctual fibre stacks on the total Hilbert cohomology by virtual correspondences, in analogy with the Fock space description in Nakajima's theory. The punctual fibre stacks are genuinely different from the symmetric stack, and has natural realizations via motivic integration.

We consider a family $(\Lambda_\lambda)_{\lambda \in \Bbbk^\times}$ of weakly symmetric algebras of Euclidean type over an algebraically closed field $\Bbbk$, where $\Bbbk^\times:= \Bbbk\setminus \{0\}$, and for each $\lambda \in \Bbbk^\times$, $\Lambda_\lambda$ is socle equivalent to $\Lambda_1$, which is the trivial extension of a hereditary algebra of type $\tilde{A}$. We compute their Hochschild cohomologies, which shows that $\Lambda_\lambda$ is not derived equivalent to $\Lambda_1$ if $\lambda \neq 1$. This means that a derived equivalence classification of weakly symmetric algebras of Euclidean type given by Bocian–Holm–Skowroński is not complete.

We extend the classical trace Paley-Wiener theorem for $p$-adic reductive groups, given by Bernstein, Deligne and Kazhdan in 1986, to a relative case. That is when $E/F$ is a quadratic extension of $p$-adic fields, the main result is a scalar Paley-Wiener theorem for relative Bessel distributions on $\mathrm{GL}_n(F)\backslash \mathrm{GL}_n(E)$. These distributions are relative character of the form $$\pi\mapsto I_\pi (f), \ f\in C_c^\infty(\mathrm{GL}_n(E)),$$ as $\pi$ ranges over $\mathrm{GL}_n(F)$-distinguished irreducible tempered representations, and are constructed from a $\mathrm{GL}_n(F)$-invariant functional and a Whittaker functional. Using the local Langlands correspondence, and the base-change from a unitary group, we show that the relative characters can be described as elements of the "generic" Bernstein center of the unitary group $U(n)$.

This paper characterizes Cohen-Macaulay algebras in the sense of Auslander and Reiten [Progr. Math. 95(1991)] within the following two classes of finite dimensional algebras: (i) truncated path algebras and (ii) Nakayama algebras. For the former, our characterization in terms of ordinary quivers shows that if a Cohen-Macaulay truncated path algebra $\Lambda$ is connected, then $\Lambda$ or $D(\Lambda)$ gives a dualizing module, where $D$ denotes the standard duality. For the latter, we show that every Cohen-Macaulay Nakayama algebra $\Lambda$ is Iwanaga-Gorenstein. In particular, $\Lambda$ always gives a dualizing module. Consequently, there is no non-trivial Cohen-Macaulay algebra among truncated path and Nakayama algebras.

In this paper, we extend Lusztig's description of quotients of Coxeter varieties by unipotent radicals of parabolic subgroups, to quotients of certain open neighbourhoods in the smooth compactification of Coxeter varieties. These quotients are products of an affine space with a Coxeter variety for an associated Levi quotient, and this result holds uniformly for any connected reductive group over a finite field. As an application, we prove a conjecture of Wang computing the invariants, under the action of unipotent subgroups, of the cohomology of the Deligne-Lusztig coverings of Coxeter varieties. We also prove that derived Alvis-Curtis duality sends the compactly supported cohomology of these coverings to their ordinary cohomology.

Let $p$ be a prime and $k$ be an algebraic closure of the finite field with $p$ elements. Let $G$ be a connected reductive algebraic group over $k$, and let $F:G\rightarrow G$ be a Frobenius morphism, defining an $\mathbb{F}_q$-rational structure on $G$, where $q$ is a power of $p$. Let $\ell$ be a prime different from $p$ and $\ell\ne 2$. Let $\Pi$ be an $\ell$-adic cuspidal unipotent representation of $G^{F^\ell}$ such that $\Pi\simeq \Pi^\sigma$, where $\sigma$ is a generator the Galois group ${\rm Gal}(\mathbb{F}_{q^\ell}/\mathbb{F}_q)$. In this paper, we prove the irreducibility of the zeroth Tate cohomology group $\widehat{\rm H}^0(\sigma,\mathcal{L})$ and vanishing of the first Tate cohomology group $\widehat{\rm H}^1(\sigma,\mathcal{L})$ for any $G^{F^\ell}\rtimes{\rm Gal}(\mathbb{F}_{q^\ell}/\mathbb{F}_q)$-stable lattice $\mathcal{L}$ in $\Pi$, using the multiplicity-one property of a progenerator constructed via the generalized Gelfand-Graev representations and Harish-Chandra induction.

Richardson varieties provide geometric representatives for products of Schubert classes, with stratifications indexed by the subintervals of Bruhat intervals. We construct analogous geometric models for tilted Bruhat intervals for all split connected reductive groups. Our construction uses Richardson varieties in affine flag varieties and yields abstract tilted Richardson varieties whose isomorphism classes, together with their labeled stratifications, are independent of the choices involved. We obtain an explicit formula for the point-counting polynomials of their open parts in terms of increasing minimal weight paths, extending Dyer's formula for Kazhdan–Lusztig $R$-polynomials. Over the real numbers, their totally nonnegative parts are regular CW complexes homeomorphic to closed balls, which gives a natural geometric realization of the posets of tilted Bruhat subintervals in all Lie types. Over the complex numbers, we prove that their fundamental classes push forward under a continuous loop evaluation map to the Poincaré duals of the minimal-degree coefficients of quantum Schubert products.

We prove that the realization functors obtained by alternating forward and backward HRS tilts of any torsion pair over a finite-dimensional algebra are equivalences from some step onward. The number of steps is bounded by the lengths of the indecomposable projective modules. The even hearts define an increasing filtration of ideals contained in a square-zero ideal. For a torsion pair induced by a two-term silting complex, we identify the successive layers with Buan-Zhou kernel ideals and compute them by reduction modulo the successive ideals. We also describe the largest quotient over which the complex becomes tilting and determine the exact stabilization index from the two parity heights. We construct algebras for which the HRS height is arbitrarily large, and compute their ideal filtrations.

The iHowe duality for iquantum groups of type AIII/IV was established in \cite{LX22} via two approaches, based on partial flag varieties and quantum coordinate coalgebras respectively. In this paper, we provide a new construction of this iHowe duality via quantum differential operators. Moreover, by deriving explicit action formulas, we prove that our construction coincides with the two constructions given in \cite{LX22}.

The TF-cone conjecture of Asai and Iyama predicts that the TF equivalence class of an integral $g$-vector is the open cone generated by the canonical summands of all its positive multiples. We give a counterexample to this conjecture. For the tensor product of the three-arrow Kronecker algebra with the path algebra of $s\to t$, a vector $w$ has a two-dimensional TF equivalence class, whereas every positive integral multiple $mw$ is generically indecomposable. The cone generated by these canonical summands is therefore a single ray. The same example disproves the relative-interior version proposed by Haerizadeh and Yassemi.

We give an explicit treatment of the third tensor power of the Clifford representation on $n$ qutrits, for $n\geq2$. This representation has five pairwise inequivalent irreducible constituents, with multiplicities $1,1,1,1,2$. The decomposition reduces the Clifford twirl of a tensor cube to four real parameters. Two of them are spectral moments of the input. The other two are cubic invariants expressed as sums of cubes of the Weyl characteristic function and the discrete Wigner function. On trace-one Hermitian matrices, the invariant polynomials of degree at most three are spanned by the constant function and four algebraically independent functions. These functions do not separate Clifford orbits. For pure inputs a single parameter suffices. We recover its range and the characterization of its maximum by stabilizer states. We bound the stabilizer fidelity, determine full separability across the three copies in the symmetric family, and recover the criterion for a complex projective $3$-design. We classify the jointly invariant quantum Boolean functions. For stabilizer testing and for testing unknown observables we give explicit bounds on copies and queries. We also give character formulas for the finite projective Clifford groups and record the complete character table for two qutrits.

Let $F$ be a $p$-adic field, let $k,c\ge1$, and let $\rho$ be a $(k,c)$ representation of $\mathrm{GL}_{kc}(F)$: a complex representation of finite length whose wave front set has $(k^c)$ as its unique maximal orbit, and which admits a $(k,c)$ functional that is unique up to scaling. We prove that the functions in the $(k,c)$ model of $\rho$ are of moderate growth, by relating this model to the Zelevinsky model of $\rho$, and obtain as a consequence a bound for the restriction of these functions to the standard Levi subgroup associated with the composition $(c^k)$ of $kc$. As applications, we study convergence problems of several integrals involving $(k,c)$ models: integrals from the generalized doubling method, and a new family of integrals which extends the Rankin–Selberg integrals for Speh representations of Lapid and Mao.

Let $k$ be a perfect field and let $A$ be a finite-dimensional representation-finite $k$-algebra. We prove that finite global dimension implies that the integral Cartan determinant of $A$ is one, extending Enomoto's result from algebraically closed fields to perfect fields.

We give new invariant-theoretic proofs of regularity for the three classical residual families DIII, balanced CII, and BDI/Spin. In DIII and balanced CII, trace contractions and a supertrace identity show that the relevant character spaces in the quadratic moment quotient vanish. These quotient-level vanishings lift to the corresponding equivariant tempered-distribution vanishings. In BDI/Spin, Gram matrices and a volume tensor give the analogous vanishing for the two single-determinant characters. We also give a direct proof of the required equivariant-distribution vanishing for EVII. These results, together with corrected distinguished-nilpotent calculations for the nice factors, establish the required componentwise regularity for the finite-component data attached to all irreducible factors and repair gaps in the previously proposed proofs. Combining these factorwise results across products and finite central quotients yields regularity for every symmetric pair of a connected complex reductive group. Generalized Harish-Chandra descent and a compatible Chevalley involution give multiplicity one for irreducible Casselman-Wallach representations in the connected-group case. A finite-component restriction argument extends this multiplicity-one statement to every possibly disconnected complex reductive $G$ whose symmetric space $G/G^\theta$ is connected. Consequently, we establish van Dijk's conjecture on the unitary Gelfand property for connected complex symmetric spaces.

We consider spin-coupled Grover walks on hypercubes within the framework of quantum Bernoulli noises (QBN). We identify parity-dependent harmonic subspaces generated by the central Hamming layers: in even dimension they give a fixed-point subspace, whereas in odd dimension the walk reduces on the harmonic subspace to an explicit eight-dimensional reduced dynamics. Using symmetric-group representation theory, we characterize the corresponding isotypic structure and determine the reduced spectrum. The harmonic reduction also yields explicit long-time probability behavior and shows that the resulting states may occupy an asymptotically vanishing fraction of the hypercube while still spanning its full Hamming diameter.

In this paper, we introduce and study the (involutive) representation homology of involutive algebras, following the development of representation homology of associative algebra in our earlier papers. Our main motivation is to understand the hyperoctahedral homology of involutive algebras. We show that over a field of characteristic $0$, the hyperoctahedral homology of an involutive algebra is isomorphic (as a graded commutative algebra) to its one dimensional involutive representation homology. This result enables the explicit computation of the hyperoctahedral homology of the universal enveloping algebras of involutive (DG) Lie algebras. For group algebras $k[\Gamma]$ equipped with the involution $g \mapsto g^{-1}$, we relate the involutive representation homology of $k[\Gamma]$ to the representation homology of the group $\Gamma$ with coefficients in $\mu_2$, and we show that the latter is the homology of the loop space of the mod $2$ Dold-Thom space of $B\Gamma$; more generally, we compute the representation homology of $\Gamma$ with coefficients in an arbitrary diagonalizable group scheme. As a consequence, the hyperoctahedral homology of $k[\Gamma]$ is concentrated in degree $0$ in characteristic $0$, for every $\Gamma$.

Let $\Phi$ be an irreducible root system normalized so that long roots have length $\sqrt{2}$, and let $\mu\leq\lambda$ be dominant weights. We associate to $(\Phi,\mu,\lambda)$ the convex polytope $$P^{\Phi}_{\mu,\lambda} = (\mu+C)\cap(\lambda-C)\cap(\overline{D}-\rho),$$ where $C=\mathbb{R}_{\geq 0}\Phi^+$ and $\overline{D}$ is the closed dominant Weyl chamber. We prove that if $P^{\Phi}_{\mu,\lambda}$ and $P^{\Phi'}_{\mu',\lambda'}$ are isometric, then $$K^{\Phi}_{\lambda\mu}(q)=K^{\Phi'}_{\lambda'\mu'}(q),$$ where $K^{\Phi}_{\lambda\mu}(q)$ is Lusztig's $q$-analogue of the $\mu$-weight multiplicity in the irreducible highest-weight representation $V(\lambda)$, often called the generalized Kostka-Foulkes polynomial. Moreover, we construct the polynomial directly from the metric shape of the polytope, and we classify when two such polytopes are isometric. Such coincidences are abundant and can occur in different Dynkin types, and in different Dynkin families. By the Lusztig-Kato theorem, the main theorem gives a metric analogue, in the spherical setting, of the combinatorial invariance conjecture for Kazhdan-Lusztig polynomials.

We construct an explicit realization of finite-dimensional irreducible representations $V_{\lambda}$ of $\mathfrak{gl}_{n}$ using the representation theory of $\mathrm{Y}(\mathfrak{gl}_{2})$. We consider $V_{\lambda}$ as the evaluation representation of $\mathrm{Y}(\mathfrak{gl}_{n})$ and restrict it to the subalgebras $\mathrm{Y}(\mathfrak{gl}_{2})^{(i)}$. Then $V_{\lambda}$ is completely reducible as a representation of $\mathrm{Y}(\mathfrak{gl}_{2})^{(i)}$. It allows us to use the explicit formulas for irreducible representations of $\mathrm{Y}(\mathfrak{gl}_{2})$.

Lazard theory is becoming increasingly important in $p$-adic representation theory, especially when studying mod $p$ representations. For $G$ a connected reductive group defined over a finite extension $K/\mathbb Q_p$, a question going back to Lazard is when a pro-$p$ Iwahori subgroup of $G(K)$ admits a saturated $p$-valuation. In this article, we establish a deep connection to Moy-Prasad theory, and show that the Moy-Prasad filtration defines a saturated $p$-valuation on $G_{x,\frac{1}{p-1}+}$ for any point $x$ in the semisimple Bruhat-Tits building of $G$, given a mild assumption on the maximal torus of $G$ and that $p>2$. We also compute Lazard Lie algebras defined over $\mathbb Z_p$ and the polynomial ring $\mathbb F_p[\varepsilon]$ associated to the saturated group $G_{x,\frac{1}{p-1}+}$, which allows us to reinterpret the Moy-Prasad isomorphism as an isomorphism of Lie algebras over $\mathbb F_p[\varepsilon]$.

We give an explicit example of a 9-dimensional 5-step nilpotent complex Lie algebra, generated by two elements, that is isomorphic to its complex conjugate but has no real form. In contrast to the earlier 10-dimensional examples, our proof is completely elementary and very short.