arXiv subjects

Toshiaki Shoji

Publications and source records attributed to Toshiaki Shoji.

35 records · Page 2Linked to original sources

arXiv2015

Double Kostka polynomials and Hall bimodule

Double Kostka polynomials are polynomials indexed by a pair of double partitions. As in the ordinary case, double Kostka polynomials are defined in terms of Schur functions and Hall-Littlewood functions associated to double partitions. In this paper, we study combinatorial properties of those double Kostka polynomials and Hall-Littlewood functions. In particular, we show that the Lascoux-Schutzenberger type formula holds for double Kostka polynomials in certain cases. Moreover, we show that the Hall bimodule introduced by Finkelberg-Ginzburg-Travkin is isomorphic to the ring of symmetric functions with two types of variables, which gives an alternate approach for their result.

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arXiv2014

Exotic symmetric space over a finite field, III

Let X be an exotic symmetric space G/H \times V, where G= GL(V) and H = Sp(V) for a symplectic vector space V over an algebraically closed field of odd characteristic. In our previous papers, character sheaves on X were constructed based on the explicit data. In this paper, we give a conceptual definition of character sheaves on X based on the idea of Ginzburg in the case of symmetric spaces, which we call Ginzburg type character sheaves. We prove that Ginzburg type character sheaves on X actually coincide with those defined in our papers. This gives a classification of Ginzburg type character sheaves on X.

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arXiv2014

Positivity properties for canonical bases of modified quantum affine $\frak{sl}_n$

The positivity property for canonical bases asserts that the structure constants of the multiplication for the canonical basis are in ${\mathbb N}[v,v^{-1}]$. Let $\mathbf U$ be the quantum group over ${\mathbb Q}(v)$ associated with a symmetric Cartan datum. The positivity property for the positive part ${\mathbf U}^+$ of ${\mathbf U}$ was proved by Lusztig. He conjectured that the positivity property holds for the modified form $\dot{\mathbf U}$ of ${\mathbf U}$. In this paper, we prove that the structure constants for the canonical basis of $\dot{\mathbf U}(\widehat{\frak{sl}}_n)$ coincide with certain structure constants for the canonical basis of ${\mathbf U}(\widehat{\frak{sl}}_N)^+$ for $n<N$. In particular, the positivity property for $\dot{\mathbf U}(\widehat{\frak{sl}}_n)$ follows from the positivity property for ${\mathbf U}(\widehat{\frak{sl}}_N)^+$.

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arXiv2014

Exotic symmetric spaces of higher level - Springer correspondence for complex reflection groups -

Let V be an 2n-dimensional vector space over an algebraically closed field of odd characteristic. Let G = GL(V), and H = Sp(V) the symplectic group contained in G. For a positive integer r > 1, we conisder the variety X = G/H \times V^{r-1}, on which H acts diagonally. X is called the exotic symmetric space of level r. Let W_{n,r} be the complex reflection group G(r,1,n). In this paper, generalizing the result for the case where r = 2, we show that there exists a natural bijective correspondence (Springer correspondence) between the set of irreducible representations of W_{n,r} and a certain set of H-equivariant simple perverse sheaves on X_{uni}$, where X_{uni} is the "unipotent part" of X. We also consider a similar problem for G \times V^{r-1}, where G = GL(V) for a finite dimensional vector space V.

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arXiv2013

Exotic symmetric space over a finite field, I

Let V be a 2n dimensional vector space over an algebraically closed field of odd characteristic. Let G = GL(V) and H = Sp_{2n} be the symplectic group contained in G. We call X = G/H \times V the exotic symmetric space, since its "unipotent" part is isomorphic to Kato's exotic nilcone. Let W be the Weyl group of type C_n. Kato established the Springer correspondence between irreducible characters of W and H-orbits in the exotic nilcone, based on the Ginzburg theory of affine Hecke algebras. In this paper, we develpoe a theory of character sheaves on X, and give an alternate proof for the Springer correspondence based on the theroy of character sheaves.

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arXiv2012

Exotic symmetric space over a finite field, II

This paper is the second part of the papers in the same title. In this paper, we prove a conjecture of Achar-Henderson, which asserts that the Poincare polynomials of the intersection cohomology complex associated to the closure of Sp_{2n}-orbits in the Kato's exotic nilpotent cone coincide with the modified Kostka polynomials indexed by double partitions, introduced by the first author. Actually this conjecture was recently proved by Kato by a different method. Our approach is based on the theory of character sheaves on the exotic symmetric space.

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arXiv2011

Iwahori's Question for Affine Hecke Algebras

In this paper we show that an affine Hecke algebra $H_q$ over complex numbers field with parameter $q\ne 1$ is not isomorphic to the group algebra over complex numbers field of the corresponding extend affine Weyl group if the corresponding root system has no factors of type $A_1$ and the order of $q$ is different from 11 and 13 if the root system has factors of type $E_8$.

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arXiv2007

Lusztig's conjecture for finite classical groups with even characteristic

The determination of scalars involved in Lusztig's conjecture for finite reductive groups $G(F_q)$ was achieved by Waldspurger in the case of symplectic groups or orthogonal groups, under the condition that $p,q$ are large enough. Here $p$ is the characteristic of the finite field $F_q$. In this paper, we determine the scalars in the case of symplectic groups with $p = 2$, by applying the theory of symmetric spaces over a finite field due to Kawanaka and Lusztig. We also obtain a partial result in the case of special orthogonal groups with $p = 2$.

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arXiv2007

Product formulas for the cyclotomic v-Schur algebra and for the canonical bases of the Fock space

In our earlier work, we have proved a product formula for certain decomposition numbers of the cyclotomic v-Schur algebra associated to the Ariki-Koike algebra. It is conjectured by Yvonne that the decomposition numbers of this algebra can be described in terms of the canonical basis of the higher level Fock space studied by Uglov. In this paper we prove a product formula related to the canonical basis of the Fock space. In view of Yvonne's conjecture, this formula is regarded as a counter-part for the Fock space of our previous formula.

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arXiv2007

Cyclotomic $q$-Schur algebras associated to the Ariki-Koike algebra

Let $S$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $H_{n,r}$ of rank $n$, introduced by Dipper-James-Mathas. For each $p = (r_1, ..., r_g)$ such that $r_1 + ... + r_g = r$, we define a subalgebra $S^p$ of $S$ and its quotient algebra $\bar S^p$. It is shown that $S^p$ is a standardly based algebra and $\bar S^p$ is a cellular algebra. By making use of these algebras, we show that certain decomposition numbers for $S$ can be expressed as a product of decomposition numbers for cyclotomic $q$-Schur algebras associated to smaller Ariki_koike algebras $H_{n_k,r_k}$.

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arXiv2006

Generalized Green functions and unipotent classes for finite reductive groups, II

In a previous paper, "Generalized Green functions and unipotent classes for finite reductive groups, I", we have determined certain unknown scalars involved in the algorithm of computing generalized Green functions in the case of SL_n. In this paper, we consider a similar problem for the case of classical groups Sp_{2n}, SO_N with arbitrary characteristic. Our result enables us, in particular, to compute Green functions in bad characteristic case, which was not known before.

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arXiv2006

A variant of the induction theorem for Springer representations

Let G be a simple algebraic group over C with the Weyl group W. For a unipotent element u of G, let B_u be the variety of Borel subgroups of G containing u. Let L be a Levi subgroup of a parabolic subgroup of G with the Weyl subgroup W_L. Assume that u is in L and let B_u^L be the corresponding variety for L. The induction theorem of Springer representations describes the W-module structure of H*(B_u) in terms of the W_L-module structure of H*(B_u^L). In this paper, we prove a certain refinement of the induction theorem by considering the action of a cyclic group of order e on H^*(B_u). As a corollary, we obtain a description of the values of Green functions at e-th root of unity.

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arXiv2005

Generalized Green functions and unipotent classes for finite reductive groups, I

The algorithm of computing generalized Green functions of a finite reductive group contains some unkonwn scalars occuring from the F_q structure of irreducible local systems on unipotent classes on G. In this paper, we determine such scalars in the case where G = SL_n with Frobenius map F of split or non-split type. In the case where F is of non-split type, we use the theory of graded Hecke algebras and their representations on the equivariant homology group due to Lusztig.

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arXiv2005

Lusztig's conjecture for finite special linear groups

In this paper, we prove Lusztig's conjecture for finite special linear groups, i.e., we show that characteristic functions of character sheaves coincide with almost characters up to scalar constants, under the condition that the characteristic of the base field is not too small. We determine those scalars explicitly. Our result gives a method of computing irreducible characters of these groups.

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arXiv2004

Subfield symmetric spaces for finite special linear groups

Let $G$ be a finite reductive group defined over a finite field $F_q$. In the case where $G$ is a special linear group, we compute the multiplicities of irreducible characters of $G(F_{q^2})$ with the character of $G(F_{q^2})$ induced from the trivial character of $G(F_q)$. We discuss the relationship between these multiplicities with the theory of Shintani descent for finite reductive groups in general. We also give some formula concerning the decomposition of the character of $G(F_{q^r})$ induced from the trivial character of $G(F_q)$ for reductive groups $G$ with any positive integer $r$.

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arXiv2002

Macdonald functions associated to complex reflection groups

Let W be the complex reflection group G(e,1,n). In the author's previous paper, Hall-Littlewood functions associated to W were introduced. In the special case where W is a Weyl group of type B_n, they are closely related to Green polynomials of finite classical groups. In this paper, we introduce a two variables version of the above Hall-Littlewood functions, as a generalization of Macdonald functions associated to symmetric groups. A generalization of Macdonald operators is also constructed, and we characterize such functions by making use of Macdonald operators, assuming a certain conjecture.

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arXiv2001

Green functions associated to complex reflection groups, II

In our previous paper, Green functions associated to complex reflection groups G(e,1,n) were discussed. It involved a combinatorial approach to the Green functions of classical groups of type B_n or C_n. In this paper, we introduce Green functions associated to complex reflection groups G(e,p,n). We define Schur functions, Hall-Littlewood functions associated to G(e,p,n), and show that our Green functions are obtained as a transition matrix between those two kinds of symmetric functions. As a special case, it gives a combinatorial treatmnet of Green functions of type D_n.

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