# A <var>φ</var><sub>1,3</sub>-Filtration of the Virasoro Minimal Series <em>M</em>(<em>p</em>, <em>p'</em>) with 1 < <em>p'</em>/<em>p</em> < 2

### Boris Feigin

Independent University of Moscow, Russian Federation### Evgeny Feigin

National Research University Higher School of Economics, Moscow, Russian Federation### M. Jimbo

University of Tokyo, Japan### Tetsuji Miwa

Kyoto University, Japan### Yoshihiro Takeyama

Graduate School of Pure and Applied Sciences, Ibaraki, Japan

## Abstract

The ﬁltration of the Virasoro minimal series representations *M__r*,*s*(*p*, *p'*) induced by the (1, 3)-primary ﬁeld φ1,3(*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 φ1,3(*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 φ1,3(*z*).

## Cite this article

Boris Feigin, Evgeny Feigin, M. Jimbo, Tetsuji Miwa, Yoshihiro Takeyama, A <var>φ</var><sub>1,3</sub>-Filtration of the Virasoro Minimal Series <em>M</em>(<em>p</em>, <em>p'</em>) with 1 < <em>p'</em>/<em>p</em> < 2. Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, pp. 213–257

DOI 10.2977/PRIMS/1210167327