Algebras and Representation Theory最新文献

Let G be a simple algebraic group and (mathcal {O}subset {mathfrak g}={mathrm {Lie,}}G) a nilpotent orbit. If H is a reductive subgroup of G with ({mathfrak h}={mathrm {Lie,}}H), then ({mathfrak g}={mathfrak h}oplus {mathfrak m}), where ({mathfrak m}={mathfrak h}^perp ). We consider the natural projections (varvec{varphi }: overline{mathcal {O}}rightarrow mathfrak {h}) and (varvec{psi }: overline{mathcal {O}}rightarrow mathfrak {m}) and two related properties of ((H, mathcal {O})):

It is shown that either of these properties implies that H is semisimple. We prove that ((mathcal {P}_1)) implies ((mathcal {P}_2)) for all (mathcal {O}) and the converse holds for (mathcal {O}_textsf{min}), the minimal nilpotent orbit. If ((mathcal {P}_1)) holds, then (varvec{varphi }) is finite and ([varvec{varphi }(e),varvec{psi }(e)]=0) for all (ein mathcal {O}). Then (overline{varvec{varphi }(mathcal {O})}) is the closure of a nilpotent H-orbit (mathcal {O}'). The orbit (mathcal {O}') is “shared” in the sense of Brylinski–Kostant (J. Am. Math. Soc. 7(2), 269–298 1994). We obtain a classification of all pairs ((H,mathcal {O})) with property ((mathcal {P}_1)) and discuss various relations between (mathcal {O}) and (mathcal {O}'). In particular, we detect an omission in the list of pairs of simple groups (H, G) having a shared orbit that was given by Brylinski and Kostant. It is also proved that ((mathcal {P}_1)) for ((H,mathcal {O}_textsf{min})) implies that (overline{G{cdot }varvec{varphi }(mathcal {O}_textsf{min})}=overline{G{cdot }varvec{psi }(mathcal {O}_textsf{min})}).

Let V and W be quiver representations over (mathbb {F}_1) and let K be a field. The scalar extensions (V^K) and (W^K) are quiver representations over K with a distinguished, very well-behaved basis. We construct a basis of ({{,textrm{Hom},}}_{KQ}(V^K,W^K)) generalising the well-known basis of the morphism spaces between string and tree modules. We use this basis to give a combinatorial characterisation of absolutely indecomposable representations. Furthermore, we show that indecomposable representations with finite nice length are absolutely indecomposable. This answers a question of Jun and Sistko.

In Fang et al. (J. Algebra 618, 67–95 2023), Fang-Lan-Xiao proved a formula about Lusztig’s induction and restriction functors which can induce Green’s formula for the path algebra of a quiver over a finite field via the trace map. In this paper, we generalize their formula to that for the mixed semisimple perverse sheaves for a quiver with an automorphism. By applying the trace map, we obtain Green’s formula for any finite-dimensional hereditary algebra over a finite field.

We present a bijection between torsion pairs in (text {coh}(mathbb )) with corresponding t-structures in (textrm{D}^{b}(text {coh}(mathbb {X}))) where (mathbb {X}) represents a weighted projective line. When focusing on the split case, we derive a bijection between this class and corresponding torsion pairs in the module category of a concealed canonical algebra. Additionally, we demonstrate that if the aisle of a split t-structure in the derived category of a hereditary category contains an Ext-projective object, then it admits a tilting complex. Finally, we use the structure of the Auslander-Reiten quiver of (textrm{D}^{b}(text {coh}(mathbb {X}))) in order to classify split t-structures in (textrm{D}^{b}(text {coh}(mathbb {X}))).

Let G be a finite group, and let (textrm{Irr}(G)) denote the set of the irreducible complex characters of G. An element (gin G) is called a vanishing element of G if there exists (chi in textrm{Irr}(G)) such that (chi (g)=0) (i.e., g is a zero of (chi )) and, in this case, the conjugacy class (g^G) of g in G is called a vanishing conjugacy class. In this paper we consider several problems concerning vanishing elements and vanishing conjugacy classes; in particular, we consider the problem of determining the least number of conjugacy classes of a finite group G such that every non-linear (chi in textrm{Irr}(G)) vanishes on one of them. We also consider the related problem of determining the minimum number of non-linear irreducible characters of a group such that two of them have a common zero.

Let (mathcal {A}) be an associative algebra containing either classical or quantum universal enveloping algebra of a semi-simple complex Lie algebra (mathfrak {g}). We present a construction of the Mickelsson algebra (Z(mathcal {A},mathfrak {g})) relative to the left ideal in (mathcal {A}) generated by positive root vectors. Our method employs a calculus on Hasse diagrams associated with classical or quantum (mathfrak {g})-modules. We give an explicit expression for a PBW basis in (Z(mathcal {A},mathfrak {g})) in the case when (mathcal {A}=U(mathfrak {a})) of a finite-dimensional Lie algebra (mathfrak {a}supset mathfrak {g}). For (mathcal {A}=U_q(mathfrak {a})) and (mathfrak {g}) the commutant of a Levi subalgebra in (mathfrak {a}), we construct a PBW basis in terms of quantum Lax operators, upon extension of the ground ring of scalars to (mathbb {C}[[hbar ]]).

We study the behavior of Dirac cohomology under Howe’s (Theta )-correspondence in the case of complex reductive dual pairs. More precisely, if ((G_1,G_2)) is a complex reductive dual pair with (G_1) and (G_2) viewed as real groups, we describe those Harish-Chandra modules (pi _1) of (G_1) with nonzero Dirac cohomology whose (Theta )-liftings (Theta (pi _1)) still have nonzero Dirac cohomology. In this case, we compute explicitly the Dirac cohomology of (Theta (pi _1)).

Given an action of a finite group on a triangulated category with a suitable strong exceptional collection, a construction of Elagin produces an associated strong exceptional collection on the equivariant category. In the untwisted case, we prove that the endomorphism algebra of the induced exceptional collection is the basic reduction of the skew group algebra of the endomorphism algebra of the original exceptional collection.

Let (varvec{widehat{G}}) be a spin group over a locally compact non-archimedean local field (varvec{F}) of odd residual characteristic. We defined lifted self-dual semisimple characters for (varvec{widehat{G}}) and from them, we constructed a large class of supercuspidal representations of (varvec{widehat{G}}) when (varvec{F}) is of characteristic zero. In this paper, we show that any positive level supercuspidal representation of (varvec{widehat{G}}) contains such a character.

Mirković–Vilonen polytopes encode in a geometrical way the numerical data present in the Kashiwara crystal (B(infty )) of a semisimple group G. We retrieve these polytopes from the coproduct of the Hopf algebra (mathscr {O}(N)) of regular functions on a maximal unipotent subgroup N of G. We bring attention to a remarkable behavior that the classical bases (dual canonical, dual semicanonical, Mirković–Vilonen) of (mathscr {O}(N)) manifest with respect to the extremal points of these polytopes, which extends the crystal operations. This study leans on a notion of stability for graded bialgebras.