In this paper, we determine $\tilde{\tau}_q\bigl(\chi^\lambda_q\bigr)$, the Gauss sums on the Iwahori-Hecke algebras of type $A$ for irreducible characters $\chi^\lambda_q$, which are $q$-analogues of those on the symmetric groups. We also explicitly determine the values of the corresponding trace function $\psi^{(n)}_q=\sum_{\lambda \vdash n} \tilde{\tau}_q\bigl(\chi^\lambda_q\bigr) \chi^\lambda_q$.

We generalize the quantum double construction of subfactors to that from arbitrary flat connections on 4-partite graphs and call it the \textit{mixed quantum double construction}. If all the four graphs of the original 4-partite graph are connected, it is easy to see that this construction produces Ocneanu's asymptotic inclusion of both subfactors generated by the original flat connection horizontally and vertically. The construction can be applied for example to the non-standard flat connections which appear in the construction of the Goodman-de la Harpe-Jones subfactors or to those obtained by the composition of flat part of any biunitary connections as in N. Sato's paper~[40]. An easy application shows that the asymptotic inclusions of the Goodman-de la Harpe-Jones subfactors are isomorphic to those of the Jones subfactors of type $A_n$ except for the cases of orbifold type. If two subfactors $A\subset B$ and $A \subset C$ have common $A$-$A$ bimodule systems, we can construct a flat connection in general. Then by applying our construction to the flat connection, we obtain the asymptotic inclusion of both $A\subset B$ and $A \subset C$. We also discuss the case when the original 4-partite graph contains disconnected graphs and give some such examples. General phenomena when disconnected graphs appear are explained by using bimodule systems.

In this paper we study some problems related to a refinement of Iwasawa theory, especially questions about the Fitting ideals of several natural Iwasawa modules and of the dual of the class groups, as a sequel to our previous papers [8], [3]. Among other things, we prove that the annihilator of $\mathbb{Z}_{p}(1)$ times the Stickelberger element is not in the Fitting ideal of the dualized Iwasawa module if the $p$-component of the bottom Galois group is elementary $p$-abelian with $p$-rank $\geq 4$. Our results can be applied to the case that the base field is $\mathbb{Q}$.

Let $G=Sp_{2r}({\mathbb C})$ be a simply connected simple algebraic group over $\mathbb{C}$ of type $C_r$, $B$ and $B_-$ its two opposite Borel subgroups, and $W$ the associated Weyl group. For $u$, $v\in W$, it is known that the coordinate ring ${\mathbb C}[G^{u,v}]$ of the double Bruhat cell $G^{u,v}=BuB\cup B_-vB_-$ is isomorphic to an upper cluster algebra $\overline{\mathcal{A}}(\textbf{i})_{{\mathbb C}}$ and the generalized minors $\Delta(k;\textbf{i})$ are the cluster variables of ${\mathbb C}[G^{u,v}]${5}. In the case $v=e$, we shall describe the generalized minor $\Delta(k;\textbf{i})$ explicitly.

Motivated by the famous Champernowne construction of a normal number, R.~Adler, M.~Keane, and M.~Smorodinsky constructed a normal number with respect to the simple continued fraction transformation. In this paper, we follow their idea and construct a normal series for the Artin continued fraction expansion in positive characteristic. A normal series for L\"uroth expansion is also discussed.

We construct double Grothendieck polynomials of classical types which are essentially equivalent to but simpler than the polynomials defined by A.~N. Kirillov in arXiv:1504.01469 and identify them with the polynomials defined by T.~Ikeda and H.~Naruse in Adv. Math. (2013) for the case of maximal Grassmannian permutations. We also give geometric interpretation of them in terms of algebraic localization map and give explicit combinatorial formulas.

The purpose of this paper is to present a condition for the power of a Coxeter element of $\mathfrak{S}_n$ to become the longest element. To be precise, given a product $C$ of $n-1$ distinct adjacent transpositions of $\mathfrak{S}_n$ in any order, we describe a condition for $C$ such that the $(n/2)$-th power $C^{n/2}$ of $C$ becomes the longest element, in terms of the Amida diagrams.

Double Kostka polynomials $K_{\boldsymbol{\lambda},\boldsymbol{\mu}}(t)$ are polynomials in $t$, indexed by double partitions $\boldsymbol{\lambda}, \boldsymbol{\mu}$. As in the ordinary case, $K_{\boldsymbol{\lambda}, \boldsymbol{\mu}}(t)$ is defined in terms of Schur functions $s_{\boldsymbol{\lambda}}(x)$ and Hall--Littlewood functions $P_{\boldsymbol{\mu}}(x;t)$. In this paper, we study combinatorial properties of $K_{\boldsymbol{\lambda},\boldsymbol{\mu}}(t)$ and $P_{\boldsymbol{\mu}}(x;t)$. In particular, we show that the Lascoux--Sch\"utzenberger type formula holds for $K_{\boldsymbol{\lambda},\boldsymbol{\mu}}(t)$ in the case where $\boldsymbol{\mu} = (-,\mu'')$. Moreover, we show that the Hall bimodule $\mathscr{M}$ introduced by Finkelberg-Ginzburg-Travkin is isomorphic to the ring of symmetric functions (with two types of variables) and the natural basis $\mathfrak{u}_{\boldsymbol{\lambda}}$ of $\mathscr{M}$ is sent to $P_{\boldsymbol{\lambda}}(x;t)$ (up to scalar) under this isomorphism. This gives an alternate approach for their result.

We prove $f_*(O_X)=O_S$ for a proper flat surjective morphism $f:X\to S$ of noetherian schemes under a mild condition.

In this paper, we classify Grothendieck dessins of X-shaped plane trees defined over the rationals.

For the fifth Painlevé equation near the origin we present two kinds of logarithmic solutions, which are represented, respectively, by convergent series with multipliers admitting asymptotic expansions in descending logarithmic powers and by those with multipliers polynomial in logarithmic powers. It is conjectured that the asymptotic multipliers are also polynomials in logarithmic powers. These solutions are constructed by iteration on certain rings of exponential type series.

We discuss the inverse Galois problem with normal basis, concerning Kummer theories for algebraic tori, in the framework of group schemes. The unit group scheme of a group algebra plays an important role in this article, as was pointed out by Serre~[8]. We develop our argument not only over a field but also over a ring, considering integral models of Kummer theories for algebraic tori.

This paper discusses a holomorphic nonlinear singular partial differential equation $(t \partial_t)^mu=F(t,x,\{(t \partial_t)^j \partial_x^{\alpha}u \}_{j+\alpha \leq m, j<m})$ that is of nonlinear totally characteristic type. The Newton Polygon at $x=0$ of the equation is defined, and by means of this polygon we define a generalized Poincaré condition (GP) and a condition (R) that the equation has a regular singularity at $x=0$. Under these conditions, (GP) and (R), it is proved that every formal power series solution is convergent in a neighborhood of the origin.

We give two alternative proofs of the invariance of the Drinfeld pairing under the action of the braid group. One uses the Shapovalov form, and the other uses a characterization of the universal $R$-matrix.

In this article, we consider an analogue of Noether's problem for the fields of cross-ratios, and discuss on a rationality problem which connects this with Noether's problem. We show that the affirmative answer of the analogue implies the affirmative answer for Noether's Problem for any permutation group with odd degree. We also obtain some negative results for various permutation groups with even degree.

For a given étale quadratic algebra $E$ over a $\mathfrak{p}$-adic field $F$, we establish a transfer of unramified test functions on the symmetric space $\mathrm{GL}(2,F)\backslash\mathrm{GL}(2,E)$ to those on a unitary hyperbolic space so that the orbital integrals match. This is an important step toward a comparison of relative trace formulas of these symmetric spaces, which would yield an example of a non-tempered analogue of a refined global Gross-Prasad conjecture.