arXiv subjects
Publications and source records attributed to Susumu Ariki.
We prove a conjecture by Dipper, James and Murphy that a bipartition is restricted if and only if it is Kleshchev. Hence the restricted bipartitions naturally label the crystal graphs of level two irreducible integrable $\mathcal{U}_v({\hat{\mathfrak{sl}}_e})$-modules and the simple modules of Hecke algebras of type $B_n$.
These are notes prepared for ICRA workshop at Torun, Poland, August 2007. In the first part, we explain results on canonical basic sets by Geck and Jacon and propose a categorification framework which is suitable for our example of Hecke algebras. In the second part, we review basics of Kashiwara crystal and explain the Fock space theory of cyclotomic Hecke algebras and its applications. In the third part, we explain Rouquier's theory of quasihereditary covers of cyclotomic Hecke algebras. We add detailed explanation of the proofs here. The third part is based on my intensive course given at Nagoya university in January 2007.
Nazarov \cite{Nazarov:brauer} introduced an infinite dimensional algebra, which he called the \textit{affine Wenzl algebra}, in his study of the Brauer algebras. In this paper we study certain ``cyclotomic quotients'' of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank $r^n(2n-1)!!$ (when $Ω$ is $\bu$--admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov--Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Weyl algebra (when $Ω$ is admissible).
We prove the modular branching rule of the cyclotomic Hecke algebras, which has remained open.
Let $W$ be a finite Weyl group of classical type which may not be irreducible, $F$ an algebraically closed field, $q$ an invertible element of $F$. We denote by $\mathcal H_W(q)$ the associated Hecke algebra. If $q=1$ then it is $FW$ and we know the representation type. Thus, we assume that $q\ne 1$. Let $P_W(x)$ be the Poincare polynomial of $W$. It is well-known that $\mathcal H_W(q)$ is semisimple if and only if $x-q$ does not divide $P_W(x)$. We show that the similar results hold for finiteness, tameness and wildness. In other words, the Poincare polynomial governs the representation type of $\mathcal H_W(q)$ completely. Note that the finiteness result was already given in the author's previous papers, some of which were written with Andrew Mathas. The proof uses the Fock space theory, which was developed for proving the LLT conjecture (see AMS Univ. Lec. Ser. 26), the Specht module theory, which was developed by Dipper, James and Murphy in this case, and results from the theory of finite dimensional algebras.
Based on a recent result of Mathas and the author, we prove that Uno's conjecture on representation types of Hecke algebras is true for all Hecke algebras of classical type.
This paper determines the representation type of the Iwahori-Hecke algebras of type B when $q\ne\pm1$. In particular, we show that a single parameter non-semisimple Iwahori-Hecke algebra of type B has finite representation type if and only if q is a simple root of the Poincaré polynomial, confirming a conjecture of Uno's.
In this short note, we describe the A_1^{(1)} soliton cellular automata as an evolution of a poset. This allows us to explain the conservation laws for the A_1^{(1)} soliton cellular automata, one given by Torii, Takahashi and Satsuma, and the other given by Fukuda, Okado and Yamada, in terms of the stack permutations of states in a very natural manner. As a biproduct, we can prove a conjectured formula relating these laws.
We give a proof of a conjecture that Kleshchev multipartitions are those partitions which parametrize non-zero simple modules obtained as factor modules of Specht modules by their own radicals.
These are lecture notes prepared for London Mathematical Society Symposium "Quantum Groups" held in Durham 19-29 July 1999. We give a survey on cyclotomic Hecke algebras. These algebras have been studied by Dipper, James, Malle, Mathas and the author. In lecture one, we start with definitions. We then go on to Specht module theory of cyclotomic Hecke algebras constructed by Dipper, James and Mathas. In lecture two, we explain Ginzburg and Lusztig's theory on affine Hecke algebras and its application to cyclotomic Hecke algebras due to the author. As Kazhdan-Lusztig bases of Hecke algebras describe decomposition numbers of quantum algebras at roots of unity, canonical bases of quantum affine algebras on integrable modules describe decomposition numbers of cyclotomic Hecke algebras at roots of unity. In particular, it turns out that crystal structure in the sense of Kashiwara exists behind the representation theory of cyclotomic Hecke algebras. This was first observed by Lascoux, Leclerc and Thibon. Its interpretation in terms of socle series was proposed by Rouquier. It is further developed in Vazirani's thesis. The special case of Hecke algebras of type A is extensively studied, and its relation to Lusztig conjecture is now very clear. We explain Varagnolo and Vasserot's work and several algorithms proposed so far. Its application to the decomposition numbers of finite general linear groups due to Dipper and James is also explained. The parametrization of simple modules of cyclotomic Hecke algebras over arbitrary fields in terms of crystal graphs of quantum affine algebras is given by Mathas and the author. In the third lecture, we show that this crystal graph description fits well in the Specht module theory. It uses work of Takemura and Uglov, Varagnolo and Vasserot.
Although the Robinson-Schensted-Knuth correspondence is a classical subject, its study is still active because of new development in last two decades. In this field, fundamental results are sometimes proved by using machineries which may be bypassed. Hence we still feel a lack of good source for these elementary materials. This paper is intended to fill in this lack about the RSK correspondence and the left cells of the symmetric group. It is mostly survey-like, but shows the variety of fields the RSK correspondence is related to.
The reduced Schur functions are studied. Their relations to the basic representation of $A^(1)_{r-1}$ and modular representations of the symmetric groups are clarified. Littlewood-Richardson coefficients appear in the linear relations among reduced Schur functions.
The basic representation of $\A$ is studied. The weight vectors are represented in terms of Schur functions. A suitable base of any weight space is given. Littlewood-Richardson rule appears in the linear relations among weight vectors.
The purpose of this paper is to give a reciprocity between $U_q(h)$ and $\Cal H_{n,r}$, the Hecke algebra of $(\Bbb Z / r\Bbb Z)\wr \frak S_n$ introduced by Ariki and Koike. Let $K=\Bbb Q(q,u_1,\dots ,u_r)$ be the field of rational funcitons in variables $q,u_1,\dots ,u_r$. We adopt $K$ as the base field for both the quantized universal enveloping algebra $U_q(gl_r)$ and the Hecke algebra $\Cal H_n$. We denote by $U_q(h)$ the $K$-subalgebra of $U_q(gl_r)$ generated by $q^{E_{ii}}\;$'s $(1\le i \le r)$. In this paper, we show that the commutant of $U_q(h)$ in $End((K^r)^{\otimes n})$ is isomorphic to a quotient of $\Cal H_{n,r}$. We also determine the irreducible decomposition of $(K^r)^{\otimes n}$ under the action of $\Cal H_{n,r}$. As a consequence, we obtain the reciprocity for $U_q(h)$ and $\Cal H_{n,r}$.