Mathematics

Let k(B_0) and l(B_0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B_0 of a finite group G. We prove that, if k(B_0)-l(B_0)=1, then l(B_0)\geq p-1 or else p=11 and l(B_0)=9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B_0)\geq p-1 or else p = 11 and G/O_{p'}(G) =11^2:SL(2,5). These results are useful in the study of principal blocks with a few characters. We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer characters in the principal p-block of G is always at least 2\sqrt{p-1}+1-k_p(G), where k_p(G) is the number of conjugacy classes of p-elements of G. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number of p-regular classes in finite groups.

Bijections between invariants associated to indecomposable projective modules over some suitable Brauer configuration algebras and invariants associated to solutions of the Kronecker problem and the four subspace problem are used to categorify integer sequences in the sense of Ringel and Fahr. Dimensions of the Brauer configuration algebras and their corresponding centers involved in the different processes are given as well.

We prove \emph{the other half} of Brauer's Height Zero Conjecture in the case of principal blocks.

Let Q be an acyclic quiver and k be an algebraically closed field. The indecomposable exceptional modules of the path algebra kQ have been widely studied. The real Schur roots of the root system associated to Q are the dimension vectors of the indecomposable exceptional modules. It has been shown in [Nájera Chávez A., Int. Math. Res. Not. 2015 (2015), 1590-1600] that for acyclic quivers, the set of positive c -vectors and the set of real Schur roots coincide. To give a diagrammatic description of c -vectors, K-H. Lee and K. Lee conjectured that for acyclic quivers, the set of c -vectors and the set of roots corresponding to non-self-crossing admissible curves are equivalent as sets [Exp. Math., to appear, arXiv:1703.09113]. In [Adv. Math. 340 (2018), 855-882], A. Felikson and P. Tumarkin proved this conjecture for 2-complete quivers. In this paper, we prove a revised version of Lee-Lee conjecture for acyclic quivers of type A , D , and E 6 and E 7 .

In this paper we deal with Calabi-Yau structures associated with (differential graded versions of) deformed multiplicative preprojective algebras, of which we provide concrete algebraic descriptions. Along the way, we prove a general result that states the existence and uniqueness of negative cyclic lifts for non-degenerate relative Hochschild classes.

The two boundary Temperley-Lieb algebra T L k arises in the transfer matrix formulation of lattice models in Statistical Mechanics, in particular in the introduction of integrable boundary terms to the six-vertex model. In this paper, we classify and study the calibrated representations---those for which all the Murphy elements (integrals) are simultaneously diagonalizable---which, in turn, corresponds to diagonalizing the transfer matrix in the associated model. Our approach is founded upon the realization of T L k as a quotient of the type C k affine Hecke algebra H k . In previous work, we studied this Hecke algebra via its presentation by braid diagrams, tensor space operators, and related combinatorial constructions. That work is directly applied herein to give a combinatorial classification and construction of all irreducible calibrated T L k -modules and explain how these modules also arise from a Schur-Weyl duality with the quantum group U q gl 2 .

An explicit formula for the canonical bilinear form on the Grothendieck ring of the Lie supergroup GL(n,m) is given. As an application we get an algorithm for the decomposition Euler characters in terms of characters of irreducible modules in the category of partially polynomial representations.

Let g be a semisimple simply-laced Lie algebra of finite type. Let C be an abelian categorical representation of the quantum group U q (g) categorifying an integrable representation V . The Artin braid group B of g acts on D b (C) by Rickard complexes, providing a triangulated equivalence ? w 0 : D b ( C μ )??D b ( C w 0 (μ) ) , where μ is a weight of V and ? w 0 is a positive lift of the longest element of the Weyl group. We prove that this equivalence is t-exact up to shift when V is isotypic, generalising a fundamental result of Chuang and Rouquier in the case g= sl 2 . For general V , we prove that ? w 0 is a perverse equivalence with respect to a Jordan-Hölder filtration of C . Using these results we construct, from the action of B on V , an action of the cactus group on the crystal of V . This recovers the cactus group action on V defined via generalised Schützenberger involutions, and provides a new connection between categorical representation theory and crystal bases. We also use these results to give new proofs of theorems of Berenstein-Zelevinsky, Rhoades, and Stembridge regarding the action of symmetric group on the Kazhdan-Lusztig basis of its Specht modules.

We equip the type A diagrammatic Hecke category with a special derivation, so that after specialization to characteristic p it becomes a p-dg category. We prove that the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. We conjecture that the p -dg Grothendieck group is isomorphic to the Iwahori-Hecke algebra, equipping it with a basis which may differ from both the Kazhdan-Lusztig basis and the p-canonical basis. More precise conjectures will be found in the sequel. Here are some other results contained in this paper. We provide an incomplete proof of the classification of all degree +2 derivations on the diagrammatic Hecke category, and a complete proof of the classification of those derivations for which the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. In particular, our special derivation is unique up to duality and equivalence. We prove that no such derivation exists in simply-laced types outside of finite and affine type A. We also examine a particular Bott-Samelson bimodule in type A_7, which is indecomposable in characteristic 2 but decomposable in all other characteristics. We prove that this Bott-Samelson bimodule admits no nontrivial fantastic filtrations in any characteristic, which is the analogue in the p-dg setting of being indecomposable.

We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite dimensional algebra, whose representation theory is analogous to blocks of Bernstein--Gelfand--Gelfand category O . When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden--Licata--Proudfoot--Webster.