Hecke algebras of finite type are cellular

Let \cH be the one-parameter Hecke algebra associated to a finite Weyl group $W$, defined over a ground ring in which ``bad'' primes for $W$ are invertible. Using deep properties of the Kazhdan--Lusztig basis of \cH and Lusztig's \ba-function, we show that \cH has a natural cellular structure in the sense of Graham and Lehrer. Thus, we obtain a general theory of ``Specht modules'' for Hecke algebras of finite type. Previously, a general cellular structure was only known to exist in types $A_n$ and $B_n$.Comment: 14 pages; added reference

Exact solutions of two complementary 1D quantum many-body systems on the half-line

We consider two particular 1D quantum many-body systems with local interactions related to the root system $C_N$. Both models describe identical particles moving on the half-line with non-trivial boundary conditions at the origin, and they are in many ways complementary to each other. We discuss the Bethe Ansatz solution for the first model where the interaction potentials are delta-functions, and we find that this provides an exact solution not only in the boson case but even for the generalized model where the particles are distinguishable. In the second model the particles have particular momentum dependent interactions, and we find that it is non-trivial and exactly solvable by Bethe Ansatz only in case the particles are fermions. This latter model has a natural physical interpretation as the non-relativistic limit of the massive Thirring model on the half-line. We establish a duality relation between the bosonic delta-interaction model and the fermionic model with local momentum dependent interactions. We also elaborate on the physical interpretation of these models. In our discussion the Yang-Baxter relations and the Reflection equation play a central role.Comment: 15 pages, a mistake corrected changing one of our conclusion

Schur elements for the Ariki-Koike algebra and applications

We study the Schur elements associated to the simple modules of the Ariki-Koike algebra. We first give a cancellation-free formula for them so that their factors can be easily read and programmed. We then study direct applications of this result. We also complete the determination of the canonical basic sets for cyclotomic Hecke algebras of type $G(l,p,n)$ in characteristic 0.Comment: The paper contains the results of arXiv:1101.146

Centers and Cocenters of 00-Hecke algebras

In this paper, we give explicit descriptions of the centers and cocenters of $0$-Hecke algebras associated to finite Coxeter groups.Comment: 13 pages, a mistake in 4.2 is correcte

Affine cellularity of affine Hecke algebras of rank two

We show that affine Hecke algebras of rank two with generic parameters are affine cellular in the sense of Koenig-Xi.Comment: 24 pages, 4 figures and 14 tables. New version: added references, corrected typos. Final versio

Hecke algebras with unequal parameters and Vogan's left cell invariants

In 1979, Vogan introduced a generalised $\\tau$ -invariant for characterising primitive ideals in enveloping algebras. Via a known dictionary this translates to an invariant of left cells in the sense of Kazhdan and Lusztig. Although it is not a complete invariant, it is extremely useful in describing left cells. Here, we propose a general framework for defining such invariants which also applies to Hecke algebras with unequal parameters.Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:1405.573

On Kazhdan-Lusztig cells in type B

32 pagesWe prove that, for any choice of parameters, the Kazhdan-Lusztig cells of a Weyl group of type $B$ are unions of combinatorial cells (defined using the domino insertion algorithm)

Enumeration of bigrassmannian permutations below a permutation in Bruhat order

In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly below a permutation in Bruhat order in the symmetric groups. For the proof, we use equivalent characterizations of bigrassmannian permutations by Lascoux-Schutzenberger and Reading.Comment: 7 pages

Estrogen Regulation of Jun and Fos in MCF-7 Cells

Abstract C-Fos and c-Jun are transcription factors that form the dimer Activator Protein 1 (AP-1) and bind DNA to initiate transcription. C-Fos, c-Jun are targets of the Extracellular Signal-Regulated Kinase (ERK) in multiple cell types, including MCF-7 breast cancer cells. The hormone estrogen (E2) can increase intracellular calcium levels which activates calcium/calmodulin-dependent kinase (CaM Kinase) proteins, which control ERK and gene transcription. Our goal was to evaluate the ability of E2 to activate c-Fos and c-Jun and induce their dimerization, via CaM KK and ERK, in MCF-7 cells. Interestingly, E2 stimulation of MCF-7 cells triggered phosphorylation of c-Jun and c-Fos an effect that was blocked with STO-609 and U0126, which target CaM KK and ERK, respectively. siRNA inhibition of CaM KK and ERK blocked E2-stimulated c-Jun and c-Fos phosphorylation. Additionally, E2 triggered AP-1 directed luciferase activity in MCF-7 cells that was blocked by inhibiting either CaM KK or ERK with siRNA. In summary, our data suggests that E2 utilizes both CaM KK and ERK to phosphorylate c-Jun and c-Fos and regulate their transcriptional activity in breast cancer cells