In this paper we study the semisimplicity of ℬ_r(C,δ ) , the Brauer algebra of type C over ℂ . In contrast to the classical Brauer algebras, which are semisimple for all but finitely many integer values of the parameter δ , the Brauer algebra of type C may fail to be semisimple even for certain nonintegral values of δ . Assuming δ 0 , we exploit the cellular structure of the algebra to analyze its cell modules and establish a criterion for semisimplicity. In particular, we prove that ℬ_r(C,δ ) is semisimple provided that δ ^2+ξδ∉ℤ for all ξ∈{1,-1} .
In this paper, we introduce the cyclic property for basic algebras and provide a necessary condition for an algebra Λ to share the same support τ -tilting poset with a given algebra Γ having . Furthermore, we show that this necessary condition is also sufficient if Γ is either a preprojective algebra of type , a Nakayama algebra, a generalized Brauer tree algebra, or a τ -tilting finite algebra with radical square zero.
Let G be a finite abelian group and let A be a G-graded algebra with color involution *. We first establish a Wedderburn–Malcev decomposition in case A is finite-dimensional. Then, we prove several results for algebras with polynomial identities in this setting, including the Hook theorem, the Specht property, and, most notably, the Representability theorems. The latter assert that A satisfies the same * -identities as the Grassmann envelope of a finite-dimensional (G×ℤ_2) -graded algebra B with a suitable color involution ♯ related to * .
Let p be an odd prime. Let G = SL_2(𝔽_p) and let B denote the subgroup of upper triangular matrices of G. Finally, let 𝔽 be an algebraically closed field of characteristic p. The Green correspondence gives a bijection between the non-projective indecomposable 𝔽[G] modules and non-projective indecomposable 𝔽[B] modules, realised by restriction and induction. In this paper, after recalling a suitable description of the non-projective indecomposable modules for these group algebras, we explicitly describe the Green correspondence bijection. We do this by pinpointing the modules’ position on the Stable Auslanden-Reiten quivers. Finally, we obtain two corollaries in terms of this description: formula for lifting the 𝔽[B] module decomposition of an 𝔽[G] module, and a complete description of Ind_B^G and Res^G_B .
Abstract Let p be an odd prime. Let $${G = SL_2(\mathbb {F}_p)}$$ G = S L 2 ( F p ) and let B denote the subgroup of upper triangular matrices of G . Finally, let $${\mathbb {F}}$$ F be an algebraically closed field of characteristic p . The Green correspondence gives a bijection between the non-projective indecomposable $${\mathbb {F}[G]}$$ F [ G ] modules and non-projective indecomposable $${\mathbb {F}[B]}$$ F [ B ] modules, realised by restriction and induction. In this paper, after recalling a suitable description of the non-projective indecomposable modules for these group algebras, we explicitly describe the Green correspondence bijection. We do this by pinpointing the modules’ position on the Stable Auslanden-Reiten quivers. Finally, we obtain two corollaries in terms of this description: formula for lifting the $${\mathbb {F}[B]}$$ F [ B ] module decomposition of an $${\mathbb {F}[G]}$$ F [ G ] module, and a complete description of $${\text { Ind}_B^G}$$ Ind B G and $${\text { Res}^G_B}$$ Res B G .
Abstract Since the establishment of the quantum Schur–Weyl duality in Jimbo (Lett. Math. Phys. 11 , 247–252, 1986), the duality pair $$(\textbf{U}(\mathfrak {gl}_n),\varvec{\mathcal {H}}(\mathfrak S_r))$$ ( U ( gl n ) , H ( S r ) ) of type A has been extended to the duality pairs $$(\textbf{U}^\jmath (n),\varvec{\mathcal {H}}(B_r))$$ ( U ȷ ( n ) , H ( B r ) ) and $$(\textbf{U}^\imath (n),\varvec{\mathcal {H}}(C_r))$$ ( U ı ( n ) , H ( C r ) ) in the Hecke algebra series in Bao and Wang (Astérisque 402 , vii+134, 2018), where $$\textbf{U}^\jmath (n),\textbf{U}^\imath (n)$$ U ȷ ( n ) , U ı ( n ) are i -quantum groups arising from certain quantum symmetric pairs. The quantum Schur algebra associated with the pair $$(\textbf{U}(\mathfrak {gl}_n),\varvec{\mathcal {H}}(\mathfrak S_r))$$ ( U ( gl n ) , H ( S r ) ) has a nice and simple presentation; see Doty and Giaquinto (2002). In this paper, we tackle the presentation problem for the i -quantum Schur algebras associated with the duality pair $$(\textbf{U}^\imath (n),\varvec{\mathcal {H}}(C_r))$$ ( U ı ( n ) , H ( C r ) ) . Such a q -Schur algebra is called the hyperoctahedral q -Schur algebras in Green (J. Algebra 192 , 418–438, 1997). See Bhattacharya (2026) for the $$\textbf{U}^\jmath (n)$$ U ȷ ( n ) case. Building on the explicit epimorphism $$\phi _{n,r}^\imath $$ ϕ n , r ı from the i -quantum group $$\textbf{U}^\imath (n)$$ U ı ( n ) to the hyperoctahedral q -Schur algebras $$\mathcal {S}^\imath (n,r)$$ S ı ( n , r ) (see Du and Wu, Pacific J. Math. 320 (1), 61–101, 2022), we compute the kernel of $$\phi _{n,r}^\imath $$ ϕ n , r ı in terms of generators. This results in a presentation for $$\mathcal {S}^\imath (n,r)$$ S ı ( n , r ) with defining relations which include not only the Doty–Giaquinto’s diagonal relations but also some tridiagonal relations.
We investigate the Galois module structure of polydifferentials for Subrao curves defined over an algebraically closed field of positive characteristic, and explicitly compute the decomposition of the space of holomorphic polydifferentials into indecomposable modules.
In this paper, we introduce the notion of generalized 𝒲 -Gorenstein modules respect to some subclass 𝒲 , extending the classical notion of Gorenstein projective modules. By exploiting the correspondence between projective modules over the endomorphism ring End_R(C) of a module C and elements of its additive closure 𝒲=Add_R(C) , we establish a fundamental correspondence between Gorenstein projective End_R(C) -modules and generalized Add_R(C) -Gorenstein modules. This result refines existing relative homological settings and provides a natural extension of well-known results in Gorenstein homological algebra. We explore key properties, such as closure under direct summands and sums, and identify conditions under which the class of generalized 𝒲 -Gorenstein modules coincides with other classes of modules, like Gorenstein projective modules.
We construct a family of extremal loop weight modules for the quantum toroidal algebra 𝒰_q(𝔤^tor) of type D. By specializing the quantum parameter q, we obtain a family of finite-dimensional modules of quantum toroidal algebras of type D at roots of unity.
Toric prevarieties are non-separated analogues of toric varieties. Perling [12] provided a combinatorial description of equivariant quasicoherent sheaves on toric varieties, extending earlier ideas of Klyachko, who outlined a general framework for equivariant torsion free sheaves in an unpublished work [9]. In this article, we present a combinatorial description of equivariant quasicoherent sheaves on toric prevarieties.
Since the establishment of the quantum Schur–Weyl duality in Jimbo (Lett. Math. Phys. 11, 247–252, 1986), the duality pair (U(𝔤𝔩_n),ℋ(𝔖_r)) of type A has been extended to the duality pairs (U^ (n),ℋ(B_r)) and (U^ (n),ℋ(C_r)) in the Hecke algebra series in Bao and Wang (Astérisque 402, vii+134, 2018), where U^ (n),U^ (n) are i-quantum groups arising from certain quantum symmetric pairs. The quantum Schur algebra associated with the pair (U(𝔤𝔩_n),ℋ(𝔖_r)) has a nice and simple presentation; see Doty and Giaquinto (2002). In this paper, we tackle the presentation problem for the i-quantum Schur algebras associated with the duality pair (U^ (n),ℋ(C_r)) . Such a q-Schur algebra is called the hyperoctahedral q-Schur algebras in Green (J. Algebra 192, 418–438, 1997). See Bhattacharya (2026) for the U^ (n) case. Building on the explicit epimorphism ϕ _n,r^ from the i-quantum group U^ (n) to the hyperoctahedral q-Schur algebras 𝒮^ (n,r) (see Du and Wu, Pacific J. Math. 320(1), 61–101, 2022), we compute the kernel of ϕ _n,r^ in terms of generators. This results in a presentation for 𝒮^ (n,r) with defining relations which include not only the Doty–Giaquinto’s diagonal relations but also some tridiagonal relations.
For a quasi-Hopf algebra H, we study two types of 1-cycle deformations for a coalgebra C within the category of Yetter-Drinfeld modules over H, _H^H𝒴D . The two deformations produce C-comodule structures in _H^H𝒴D and new coalgebra structures on C in _H^H𝒴D , respectively. We show that the isomorphism types of these structures are described by a 1-homology ℋ^1_H(C, H_0) that we will introduce. Then we apply our results to the so called symplectic fermion quasi-Hopf algebras, algebras recently introduced by Farsad, Gainutdinov and Runkel.
Let n∈ℤ^≥2, ℓ∈ℤ^≥1 . In this paper we use (some slightly modified versions of) the distinguished bases {ℬ_𝔰𝔱} and {ℬ̌_𝔰𝔱} of the cyclotomic Hecke algebra ℋ_ℓ ,n of type G(ℓ ,1,n) introduced by Mathas and the first named author (Hu and Mathas, A. Math. Ann. 364, 1189–1254, 2016) to study the alternating cyclotomic Hecke algebra ℋ_ℓ ,n^# . We construct an explicit integral basis for the alternating cyclotomic Hecke algebra ℋ_ℓ ,n^# of arbitrary higher levels. We also present an explicit seminormal basis for the semisimple alternating cyclotomic Hecke algebra. We show that the alternating cyclotomic Hecke algebra is a symmetric algebra over an infinite field.
Let G be a finite group. We investigate the cohomology ring H^*(G,b;A) of a block ideal b of the finite group algebra kG, where k is an algebraically close field of characteristic p and A is a source algebra of b. The author expected in Sasaki (Algebr. Represent. Theory 16, 1039–1049, 2013) that, P being its defect group, the image of the transfer map t_A of the cohomology ring H^*(P,k) induced by A coincides with H^*(G,b;A) : Im t_A=H^*(G,b;A) . This expectation has previously been settled in Sasaki (Hokkaido Math. J. 53, 443–462, 2024) for blocks of tame representation type and blocks with extraspecial p-groups as P. In this note, we prove that this expectation holds for b whose defect group P is isomorphic to a wreathed 2-group. The proof relies on prior works Okuyama and Sasaki (Algebr. Represent. Theory 4, 405–444, 2001; J. Algebra 497, 92–101, 2018), Kawai and Sasaki (J. Algebra 306(2), 301–321, 2006), and Sasaki (J. Algebra 666, 777–793, 2025).
In this article, we present a combinatorial formula for computing the Wedderburn decomposition of the rational group algebra associated with an ordinary metacyclic p-group G, where p is any prime. We also provide a formula for counting irreducible rational representations of G with distinct degrees and derive a method to explicitly construct all inequivalent irreducible rational matrix representations of G.
This paper gives an algebraic presentation of an algebra called the fused permutations algebra in the one-boundary case. It is obtained through a detailed study of the cyclotomic degenerate affine Hecke algebra. In particular, we prove that the fused permutations algebra is a quotient of the cyclotomic degenerate affine Hecke algebra, and we also describe a basis combinatorially in terms of signed permutations with avoiding patterns. In order to understand this quotient, we study the primitive idempotents of this cyclotomic degenerate affine Hecke algebra.
We answer a question raised by Auslander and Bridger by showing that not every 2-reflexive module is reflexive.
Let be an algebraically closed field of characteristic 2 and let 𝔣𝔰𝔩(2) be the unique, up to isomorphism, 3-dimensional simple Lie algebra over . Denote by 𝔪 the minimal 2-envelope of 𝔣𝔰𝔩(2) and by 𝔲(𝔪) its corresponding restricted enveloping algebra. The non-isomorphic finite-dimensional indecomposable 𝔲(𝔪) -modules were classified in [1]. In this paper, the Green ring (or representation ring) for 𝔲(𝔪) is calculated. Also, the semisimplification of the representation category of 𝔲(𝔪) is determined.
For a complex simple Lie algebra $\mathfrak{g}$ or rank $r$, let $\rho$ be the half sum of positive roots and $P(2\rho)\subset \mathbb{R}^r$ be the convex hull of all dominant weights $\lambda$ of the form $\lambda=2\rho-\sum_{i=1}^r a_i\alpha_i$ with $a_i\in \mathbb{Z}_{\geq 0}$ for $1\leq i\leq r$. We prove that if $\lambda$ is a vertex of $P(2\rho)$, then $V(\lambda)$ appears in $V(\rho) \otimes V(\rho)$ with multiplicity one. This result allows us to give an alternative proof for a weaker form (up to saturation factor) of a conjecture of Kostant that describes the components of $V(\rho)\otimes V(\rho)$. Further, using works of Knutson-Tau on the saturation property of $\mathfrak{sl_{r+1}}$, our results give an alternative proof of Kostant's conjecture in the particular case $\mathfrak{g}=\mathfrak{sl_{r+1}}$.
