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Paths, Crystals and Fermionic Formulae

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Book cover MathPhys Odyssey 2001

Part of the book series: Progress in Mathematical Physics ((PMP,volume 23))

Abstract

We introduce a fermionic formula associated with any quantum affine algebra U q (X (r) N . Guided by the interplay between corner transfer matrix and the Bethe ansatz in solvable lattice models, we study several aspects related to representation theory, most crucially, the crystal basis theory. They include one-dimensional sums over both finite and semi-infinite paths, spinon character formulae, Lepowsky—Primc type conjectural formula for vacuum string functions, dilogarithm identities, Q-systems and their solution by characters of various classical subalgebras and so forth. The results expand [HKOTY1] including the twisted cases and more details on inhomogeneous paths consisting of non-perfect crystals. As a most intriguing example, certain inhomogeneous one-dimensional sums conjecturally give rise to branching functions of an integrable G (1)2 -module related to the embedding G (1)2 ↪ B (1)3 ↪ D 14 .

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Authors and Affiliations

  1. Institute of Physics, University of Tokyo, Komaba, Tokyo, 153-8902, Japan

    Goro Hatayama & Atsuo Kuniba

  2. Department of Informatics and Mathematical Science Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, 560-8531, Japan

    Masato Okado

  3. Department of Applied Physics, National Defense Academy, Yokosuka, 239-8686, Japan

    Taichiro Takagi

  4. Graduate School of Mathematical Sciences, University of Tokyo, Tokyo, 153-8914, Japan

    Zengo Tsuboi

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  1. Goro Hatayama
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  2. Atsuo Kuniba
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  3. Masato Okado
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  4. Taichiro Takagi
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  5. Zengo Tsuboi
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Editors and Affiliations

  1. RIMS, Kyoto University, Kyoto, 606-8502, Japan

    Masaki Kashiwara

  2. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Tetsuji Miwa

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Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Tsuboi, Z. (2002). Paths, Crystals and Fermionic Formulae. In: Kashiwara, M., Miwa, T. (eds) MathPhys Odyssey 2001. Progress in Mathematical Physics, vol 23. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0087-1_9

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