# Publications of the Research Institute for Mathematical Sciences

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**Volume 44, Issue 2, 2008, pp. 213–257**

**DOI: 10.2977/prims/1210167327**

A `φ`_{1,3}-Filtration of the Virasoro Minimal Series *M*(*p*, *p'*) with 1 < *p'*/*p* < 2

^{[1]}, Boris Feigin

^{[2]}, M. Jimbo

^{[3]}, Tetsuji Miwa

^{[4]}and Y. Takeyama

^{[5]}(1) Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, 119002, MOSCOW, RUSSIAN FEDERATION

(2) Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, 119002, MOSCOW, RUSSIAN FEDERATION

(3) Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, 153-8914, TOKYO, JAPAN

(4) Department of Mathematics, Kyoto University, 606-8502, KYOTO, JAPAN

(5) Institute of Mathematics, Graduate School of Pure and Applied Sciences, University of Tsukuba, 305-8571, IBARAKI, TSUKUBA, JAPAN

The ﬁltration of the Virasoro minimal series representations *M*_{r,s}^{(p, p')} induced by the (1, 3)-primary ﬁeld `φ`_{1,3}(*z*) is studied. For 1 < *p'*/*p* < 2, a conjectural basis of *M*_{r,s}^{(p, p')} compatible with the ﬁltration is given by using monomial vectors in terms of the Fourier coeffcients of `φ`_{1,3}(*z*). In support of this conjecture, we give two results. First, we establish the equality of the character of the conjectural basis vectors with the character of the whole representation space. Second, for the unitary series (*p'* = *p* + 1), we establish for each *m* the equality between the character of the degree *m* monomial basis and the character of the degree *m* component in the associated graded module gr(*M*_{r,s}^{(p, p+1)}) with respect to the ﬁltration deﬁned by `φ`_{1,3}(*z*).

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Feigin E, Feigin B, Jimbo M, Miwa T, Takeyama Y. A `φ`_{1,3}-Filtration of the Virasoro Minimal Series *M*(*p*, *p'*) with 1 < *p'*/*p* < 2. *Publ. Res. Inst. Math. Sci.* 44 (2008), 213-257. doi: 10.2977/prims/1210167327