arXiv subjects
Publications and source records attributed to Susumu Ariki.
We determine the representation type of cyclotomic quiver Hecke algebras of affine type C. In the tame cases, we explicitly describe their basic algebras under the assumption $\text{ch}\ \mathbb{k}\ne2$, relying on the Morita invariance of cellularity.
Schurian-finiteness, also known as $τ$-tilting finiteness, is equivalent to the finiteness of various representation theoretic objects such as wide subcategories. The first three authors classified Schurian-finite blocks of type A Hecke algebras in [ALS23]. Here we study the Schurian-finiteness of blocks of type B Hecke algebras, and determine the Schurian-finiteness of all blocks if the Hecke algebra is `non-integral', and for almost all blocks in the integral case. The only remaining cases are a small number of blocks in defect $3$ when $e=3$, and a family of blocks in defects $3$ and $4$ for $e\geqslant4$. The classification is mostly achieved by methods using decomposition numbers, with many degenerate cases requiring direct study using standard methods from the representation theory of quivers.
We first investigate a connected quiver consisting of all dominant maximal weights for an integrable highest weight module in affine type A. This quiver provides an efficient method to obtain all dominant maximal weights. Then, we completely determine the representation type of cyclotomic Khovanov-Lauda-Rouquier algebras of arbitrary level in affine type A, by using the quiver we construct. This result gives a complete classification for the representation type of blocks of cyclotomic Hecke algebras since cyclotomic KLR algebras of type $A^{(1)}_\ell$ form a one-parameter family and cyclotomic Hecke algebras occur at a special parameter, i.e., $t=-2$ if $\ell=1$ and $t=(-1)^{\ell+1}$ if $\ell\geq2$.
For any algebra $A$ over an algebraically closed field $\mathbb{F}$, we say that an $A$-module $M$ is Schurian if $\mathrm{End}_A(M) \cong \mathbb{F}$. We say that $A$ is Schurian-finite if there are only finitely many isomorphism classes of Schurian $A$-modules, and Schurian-infinite otherwise. By work of Demonet, Iyama and Jasso it is known that Schurian-finiteness is equivalent to $τ$-tilting-finiteness, so that we may draw on a wealth of known results in the subject. We prove that for the type $A$ Hecke algebras with quantum characteristic $e\geq 3$, all blocks of weight at least $2$ are Schurian-infinite in any characteristic. Weight $0$ and $1$ blocks are known by results of Erdmann and Nakano to be representation finite, and are therefore Schurian-finite. This means that blocks of type $A$ Hecke algebras (when $e\geq 3$) are Schurian-infinite if and only if they have wild representation type if and only if the module category has finitely many wide subcategories. Along the way, we also prove a graded version of the Scopes equivalence, which is likely to be of independent interest.
We classify tame block algebras of Hecke algebras of classical type over an algebraically closed field of odd characteristic.
We construct and investigate Specht modules $\mathcal{S}^λ$ for cyclotomic quiver Hecke algebras in type $C^{(1)}_\ell$ and $C_\infty$, which are labelled by multipartitions $λ$. It is shown that in type $C_\infty$, the Specht module $\mathcal{S}^λ$ has a homogeneous basis indexed by standard tableaux of shape $λ$, which yields a graded character formula and good properties with the exact functors $E_i^Λ$ and $F_i^Λ$. For type $C^{(1)}_\ell$, we propose a conjecture.
Let $A$ be a truncated polynomial ring over a complete discrete valuation ring $\mathcal{O}$, and we consider the additive category consisting of $A$-lattices $M$ with the property that $M\otimes \mathcal{K}$ is projective as an $A\otimes \mathcal{K}$-module, where $\mathcal{K}$ is the fraction field of $\mathcal{O}$. Then, we may define the stable Auslander-Reiten quiver of the category. We determine the shape of the components of the stable Auslander-Reiten quiver that contain Heller lattices.
We classify Morita equivalence classes of indecomposable self-injective cellular algebras which have polynomial growth representation type, assuming that the base field has an odd characteristic. This assumption on the characteristic is for the cellularity to be a Morita invariant property.
We find representation type of the cyclotomic quiver Hecke algebras of level two in affine type A. In particular, we have determined representation type for all the block algebras of Hecke algebras of classical type (except for characteristic two in type D), which has not been known for a long time. As an application of this result, we prove that block algebras of finite representation type are Brauer tree algebras whose Brauer trees are straight lines without exceptional vertex if the Hecke algebras are of classical type and the characteristic of the base field is odd. We conjecture that this statement should hold for Hecke algebras of exceptional type with bad primes invertible in the base field.
We give Erdmann-Nakano type theorem for the finite quiver Hecke algebras $R^{Λ_0}(β)$ of affine type $D^{(2)}_{\ell+1}$, which tells their representation type. If $R^{Λ_0}(β)$ is not of wild representation type, we may compute its stable Auslander-Reiten quiver.
We give a graded dimension formula described in terms of combinatorics of Young diagrams and a simple criterion to determine the representation type for the finite quiver Hecke algebras of type $C_{\ell}^{(1)}$.
We give Erdmann-Nakano type theorem for the finite quiver Hecke algebras $R^{Λ_0}(β)$ of affine type $A^{(1)}_{\ell}$. Note that each finite quiver Hecke algebra lies in one parameter family, and the original Erdmann-Nakano theorem studied the finite quiver Hecke algebra at a special parameter value. We study the general case in our paper. Our result shows in particular that their representation type does not depend on the parameter. Moreover, when the parameter value is nonzero, we show that finite quiver Hecke algebras of tame representation type are biserial algebras.
We study cyclotomic quiver Hecke algebras $R^{Λ_0}(β)$ in type $A^{(2)}_{2\ell}$, where $Λ_0$ is the fundamental weight. The algebras are natural $A^{(2)}_{2\ell}$-type analogue of Iwahori-Hecke algebras associated with the symmetric group, from the viewpoint of the Fock space theory developed by the first author and his collaborators. We give a formula for the dimension of the algebra, and a simple criterion to tell the representation type. The criterion is a natural generalization of Erdmann and Nakano's for the Iwahori-Hecke algebras. Except for the examples coming from cyclotomic Hecke algebras, no results of these kind existed for cyclotomic quiver Hecke algebras, and our results are the first instances beyond the case of cyclotomic Hecke algebras.
Generalizing recent work of Brundan and Kleshchev, we introduce grading on Dipper-James' $q$-Schur algebra, and prove a graded analogue of the Leclerc and Thibon's conjecture on the decomposition numbers of the $q$-Schur algebra when $q^2\neq1$ and $q^3\neq1$.
The level l Fock space admits canonical bases G_e and G_\infty. They correspond to U_{v}(hat{sl}_{e}) and U_{v}(sl_{\infty})-module structures. We establish that the transition matrices relating these two bases are unitriangular with coefficients in N[v]. Restriction to the highest weight modules generated by the empty l-partition then gives a natural quantization of a theorem by Geck and Rouquier on the factorization of decomposition matrices which are associated to Ariki-Koike algebras.
For the affine Hecke algebra of type A at roots of unity, we make explicit the correspondence between geometrically constructed simple modules and combinatorially constructed simple modules and prove the modular branching rule. The latter generalizes work by Vazirani.
For the affine Hecke algebra of type A at roots of unity, we make explicit the correspondence between geometrically constructed simple modules and combinatorially constructed simple modules and prove the modular branching rule. The latter generalizes work by Vazirani.
Let $\{B(Λ_m)|m\in\Z/e\Z\}$ be the set of level one $\mathfrak{g}(A^{(1)}_{e-1})$-crystals, and consider the realization of $B(Λ_m)$ using $e$-restricted partitions. We prove a purely Young diagrammatic criterion for an element of $B(Λ_0)^{\otimes d_1}\otimes B(Λ_m)^{\otimes d_2}$ to be in the component $B(d_1Λ_0+d_2Λ_m)$. As an application, we give a non-recursive characterization of simple modules of the Hecke algebra of type $B$. In the course of the proof, we also obtain a combinatorial description of the second type of Kashiwara's Demazure crystal in $B(Λ_m)$.