# Part I, Symplectic ice, Part II, Global and local Kubota symbols

## Abstract/Contents

- Abstract
- My Ph.D. thesis consists of two parts. In the first part I construct an ice-model (a six-vertex model). I show the partition function of this ice model is equal to the product of an irreducible character of the symplectic group and a deformation of the Weyl denominator. A similar result was originally proved by Hamel and King, but the Boltzmann weights (for the vertices at the caps) that I use are different then the ones that are used by Hamel and King. Also, my proof of this result uses Yang-Baxter equation (while the proof of Hamel and King does not). This gives us a 6-vertex models for characters of the symplectic group , but this result can also be interpreted as an example of an exactly solved model, that is, an ice model whose partition function can be computed explicitly (this is of interest to people who work in statistical mechanics). In the second part of my thesis I work with the global and local Kubota symbols. I introduce a local Kubota symbol, and show that it satisfies some properties that are satisfies by the global Kubota symbol; in particular, it satisfies the reciprocity law. Using the parallel between the properties of the global and local Kubota symbols, I give a new proof of Kubota's theorem (for the case when p does not divide n), and then generalize this theorem to the case when p divides n. In the last chapter of my thesis I work with the global Kubota symbol. I investigate the levels for the global Kubota symbols for particular fields, and I obtain new levels for these fields which improve the results of Kubota, and of Bass, Milnor and Serre.

## Description

Alternative title | Symplectic ice |
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Alternative title | Global and local Kubota symbols |

Type of resource | text |

Form | electronic; electronic resource; remote |

Extent | 1 online resource. |

Publication date | 2010 |

Issuance | monographic |

Language | English |

## Creators/Contributors

Associated with | Ivanov, Dmitriy |
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Associated with | Stanford University, Department of Mathematics |

Primary advisor | Bump, Daniel, 1952- |

Thesis advisor | Bump, Daniel, 1952- |

Thesis advisor | Licata, Anthony M |

Thesis advisor | Venkatesh, Akshay, 1981- |

Advisor | Licata, Anthony M |

Advisor | Venkatesh, Akshay, 1981- |

## Subjects

Genre | Theses |
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## Bibliographic information

Statement of responsibility | Dmitriy Ivanov. |
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Note | Submitted to the Department of Mathematics. |

Thesis | Ph.D. Stanford University 2010 |

Location | electronic resource |

## Access conditions

- Copyright
- © 2010 by Dmitriy Ivanov
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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