arXiv subjects

Masaki Watanabe

Publications and source records attributed to Masaki Watanabe.

9 recordsLinked to original sources

arXiv2024

Schur partition theorems via perfect crystal

Motivated by spin modular representations of the symmetric groups, we propose two generalizations of the Schur regular partitions for an odd integer $p\geq 3$. One forms a subset of the set of $p$-strict partitions, and the other forms that of strict partitions. We prove that each set has a basic $A^{(2)}_{p-1}$-crystal structure. For $p=3$, it reproves Schur's 1926 partition theorem, a mod 6 analog of Rogers-Ramanujan partition theorem (RRPT). For $p=5$, it gives a computer-free proof of a conjecture by Andrews during his 3-parameter generalization of RRPT, which was first proved by Andrews-Bessenrodt-Olsson.

math.QA↗

arXiv2020

Reflection positivity and complex analysis of the Yang-Mills theory from a viewpoint of gluon confinement

In order to understand the confining decoupling solution of the Yang-Mills theory in the Landau gauge, we consider the massive Yang-Mills model which is defined by just adding a gluon mass term to the Yang-Mills theory with the Lorentz-covariant gauge fixing term and the associated Faddeev-Popov ghost term. First of all, we show that massive Yang-Mills model is obtained as a gauge-fixed version of the gauge-invariantly extended theory which is identified with the gauge-scalar model with a single fixed-modulus scalar field in the fundamental representation of the gauge group. This equivalence is obtained through the gauge-independent description of the Brout-Englert-Higgs mechanism proposed recently by one of the authors. Then, we reconfirm that the Euclidean gluon and ghost propagators in the Landau gauge obtained by numerical simulations on the lattice are reproduced with good accuracy from the massive Yang-Mills model by taking into account one-loop quantum corrections. Moreover, we demonstrate in a numerical way that the Schwinger function calculated from the gluon propagator in the Euclidean region exhibits violation of the reflection positivity at the physical point of the parameters. In addition, we perform the analytic continuation of the gluon propagator from the Euclidean region to the complex momentum plane towards the Minkowski region. We give an analytical proof that the reflection positivity is violated for any choice of the parameters in the massive Yang-Mills model, due to the existence of a pair of complex conjugate poles and the negativity of the spectral function for the gluon propagator to one-loop order. The complex structure of the propagator enables us to explain why the gluon propagator in the Euclidean region is well described by the Gribov-Stingl form.

hep-th↗

arXiv2019

Complex poles, spectral function and reflection positivity violation of Yang-Mills theory

We discuss the analytic continuation of the gluon propagator from the Euclidean region to the complex squared-momentum plane towards the Minkowski region from a viewpoint of gluon confinement. For this purpose, we investigate the massive Yang-Mills model with one-loop quantum corrections, which is to be identified with a low-energy effective theory of the Yang-Mills theory in the sense that the confining decoupling solution for the Euclidean gluon and ghost propagators of the Yang-Mills theory in the Landau gauge obtained by numerical simulations on the lattice are reproduced with good accuracy from the massive Yang-Mills model by taking into account one-loop quantum corrections. We show that the gluon propagator in the massive Yang-Mills model has a pair of complex conjugate poles or "tachyonic" poles of multiplicity two, in accordance with the fact that the gluon field has a negative spectral function, while the ghost propagator has at most one "unphysical" pole. These results are consistent with general relationships between the number of complex poles of a propagator and the sign of the spectral function originating from the branch cut in the Minkowski region under some assumptions on the asymptotic behaviors of the propagator. Consequently, we give an analytical proof for violation of the reflection positivity as a necessary condition for gluon confinement for any choice of the parameters in the massive Yang-Mills model, including the physical point. Moreover, the complex structure of the propagator enables us to explain why the gluon propagator in the Euclidean region is well described by the Gribov-Stingl form.

hep-th↗

arXiv2016

Kraśkiewicz-Pragacz modules and Pieri and dual Pieri rules for Schubert polynomials

In their 1987 paper Kraśkiewicz and Pragacz defined certain modules, which we call KP modules, over the upper triangular Lie algebra whose characters are Schubert polynomials. In a previous work the author showed that the tensor product of KP modules always has a KP filtration, i.e. a filtration whose each successive quotients are isomorphic to KP modules. In this paper we explicitly construct such filtrations for certain special cases of these tensor product modules, namely $\mathcal{S}_w \otimes S^d(K^i)$ and $\mathcal{S}_w \otimes \bigwedge^d(K^i)$, corresponding to Pieri and dual Pieri rules for Schubert polynomials.

math.RT↗

arXiv2015

Kraśkiewicz-Pragacz modules and Ringel duality

Kraśkiewicz and Pragacz introduced representations of the upper-triangular Lie algebras whose characters are Schubert polynomials. In a previous work the author studied the structure of Kraśkiewicz-Pragacz modules using the theory of highest weight categories. From the results there, in particular we obtain a certain highest weight category whose standard modules are KP modules. In this paper we show that this highest weight category is self Ringel-dual: this leads to an interesting symmetry relation on Ext groups between KP modules. We also show that the tensor product operation on b-modules is compatible with Ringel duality functor.

math.RT↗

arXiv2015

Pattern avoidance seen in multiplicities of maximal weights of affine Lie algebra representations

We prove that the multiplicities of certain maximal weights of $\mathfrak{g}(A^{(1)}_{n})$-modules are counted by pattern avoidance on words. This proves and generalizes a conjecture of Misra-Rebecca. We also prove similar phenomena in types $A^{(2)}_{2n}$ and $D^{(2)}_{n+1}$. Both proofs are applications of Kashiwara's crystal theory.

math.RT↗

arXiv2015

An approach toward Schubert positivities of polynomials using Kraśkiewicz-Pragacz modules

In this paper, we investigate properties of modules introduced by Kraśkiewicz and Pragacz which realize Schubert polynomials as their characters. In particular, we give some characterizations of modules having a filtration by Kraśkiewicz-Pragacz modules. In finding criteria for filtrations, we calculate generating sets for the annihilator ideals of the lowest vectors in Kraśkiewicz-Pragacz modules, and derive a projectivity result concerning Kraśkiewicz-Pragacz modules.

math.RT↗

arXiv2014

Tensor product of Kraśkiewicz and Pragacz's modules

This paper explores further properties of modules related with Schubert polynomials, introduced by Kraśkiewicz and Pragacz. In this paper we show that any tensor product of Kraśkiewicz-Pragacz modules admits a filtration by Kraśkiewicz-Pragacz modules. This result can be seen as a module-theoretic counterpart of a classical result that the product of Schubert polynomials is a positive sum of Schubert polynomials.

math.RT↗

arXiv2013

On a relation between certain character values of symmetric groups and its connection with creation operators of symmetric functions

In this paper, we derive a relation of new kind between certain character values of symmetric groups in terms of so-called maya diagrams. We also investigate a relation between our result and Bernstein's creation operators for Schur functions, and consider analogous relations for projective characters of symmetric groups through creation operators for Schur $Q$-functions. We also consider analogous relations for characters of Brauer algebras and walled Brauer algebras.

math.RT↗