Mostrando recursos 1 - 20 de 1.338

  1. Orbital Integrals on Unitary Hyperbolic Spaces Over $\frak p$-adic Fields

    TSUZUKI, Masao
    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.

  2. Toward Noether's Problem for the Fields of Cross-ratios

    TSUNOGAI, Hiroshi
    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.

  3. Invariance of the Drinfeld Pairing of a Quantum Group

    TANISAKI, Toshiyuki
    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.

  4. Generalized Poincaré Condition and Convergence of Formal Solutions of Some Nonlinear Totally Characteristic Equations

    TAHARA, Hidetoshi
    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

  5. Kummer Theories for Algebraic Tori and Normal Basis Problem

    SUWA, Noriyuki
    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.

  6. Logarithmic Solutions of the Fifth Painlevé Equation near the Origin

    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.

  7. On Mono-nodal Trees and Genus One Dessins of Pakovich-Zapponi Type

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

  8. The Direct Image Sheaf $f_*(O_X)$

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

  9. Double Kostka polynomials and Hall bimodule

    LIU, Shiyuan; SHOJI, Toshiaki
    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...

  10. Coxeter Elements of the Symmetric Groups Whose Powers Afford the Longest Elements

    KOSUDA, Masashi
    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.

  11. Construction of Double Grothendieck Polynomials of Classical Types using IdCoxeter Algebras

    KIRILLOV, Anatol N.; NARUSE, Hiroshi
    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.

  12. On the Construction of Continued Fraction Normal Series in Positive Characteristic

    KIM, Dong Han; NAKADA, Hitoshi; NATSUI, Rie
    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.

  13. Explicit Forms of Cluster Variables on Double Bruhat Cells $G^{u,e}$ of Type C

    KANAKUBO, Yuki; NAKASHIMA, Toshiki
    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.

  14. Fitting Ideals of Iwasawa Modules and of the Dual of Class Groups

    GREITHER, Cornelius; KURIHARA, Masato
    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 $\mathbb{Q}$.

  15. Mixed Quantum Double Construction of Subfactors

    GOTO, Satoshi
    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...

  16. Gauss Sums on the Iwahori-Hecke Algebras of Type $A$

    GOMI, Yasushi
    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 $\psi^{(n)}_q=\sum_{\lambda \vdash n} \tilde{\tau}_q\bigl(\chi^\lambda_q\bigr) \chi^\lambda_q$.

  17. On the Moduli Space of Pointed Algebraic Curves of Low Genus III ---Positive Characteristic---

    NAKANO, Tetsuo
    In his classical work, Pinkham discovered a beautiful theorem on the moduli space of pointed algebraic curves with a fixed Weierstrass gap sequence at the marked point. Namely, the complement of a Weierstrass gap sequence in the set of non-negative integers is a numerical semigroup, and he described such a moduli space in terms of the negative part of the miniversal deformation space of the monomial curve of this semigroup. Unfortunately, his theorem holds only in characteristic 0 and does not hold in positive characteristic in general. In this paper, we will study his theorem in positive characteristic, and give a fairly sharp...

  18. A Sufficient Condition for Orbits of Hermann Actions to be Weakly Reflective

    OHNO, Shinji
    In this paper, we give sufficient conditions for orbits of Hermann actions to be weakly reflective in terms of symmetric triads, that is a generalization of irreducible root systems. Using these sufficient conditions, we obtain new examples of weakly reflective submanifolds in compact symmetric spaces.

  19. Asymptotically Unweighted Shifts, Hypercyclicity, and Linear Chaos

    NATSUME, Ayuko; TANIGUCHI, Masahiko
    In this paper, we introduce weighted backward shifts, which are asymptotically unweighted, and give several conditions for such operators on the classical $\ell^p$ spaces to be hypercyclic and chaotic.

  20. Reidemeister Torsion and Dehn Surgery on Twist Knots

    TRAN, Anh T.
    We compute the Reidemeister torsion of the complement of a twist knot in $S^3$ and that of the 3-manifold obtained by a $\frac{1}{q}$-Dehn surgery on a twist knot.

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