Fermionic expressions for minimal model Virasoro characters

Fermionic expressions for all minimal model Virasoro characters $\chi^{p, p'}_{r, s}$ are stated and proved. Each such expression is a sum of terms of fundamental fermionic form type. In most cases, all these terms are written down using certain trees which are constructed for $s$ and $r$ from the Takahashi lengths and truncated Takahashi lengths associated with the continued fraction of $p'/p$. In the remaining cases, in addition to such terms, the fermionic expression for $\chi^{p, p'}_{r, s}$ contains a different character $\chi^{\hat p, \hat p'}_{\hat r,\hat s}$, and is thus recursive in nature. Bosonic-fermionic $q$-series identities for all characters $\chi^{p, p'}_{r, s}$ result from equating these fermionic expressions with known bosonic expressions. In the cases for which $p=2r$, $p=3r$, $p'=2s$ or $p'=3s$, Rogers-Ramanujan type identities result from equating these fermionic expressions with known product expressions for $\chi^{p, p'}_{r, s}$. The fermionic expressions are proved by first obtaining fermionic expressions for the generating functions $\chi^{p, p'}_{a, b, c}(L)$ of length $L$ Forrester-Baxter paths, using various combinatorial transforms. In the $L\to\infty$ limit, the fermionic expressions for $\chi^{p, p'}_{r, s}$ emerge after mapping between the trees that are constructed for $b$ and $r$ from the Takahashi and truncated Takahashi lengths respectively.Comment: 153 pages, includes eps figures. v2: exceptional cases clarified, (1.45/6) corrected for d=1, Section 7.5 rewritten, reference added. v3: minor typos and clarifications. To appear in Memoirs of the American Mathematical Societ

A quartet of fermionic expressions for M(k,2k±1)M(k,2k\pm1) Virasoro characters via half-lattice paths

We derive new fermionic expressions for the characters of the Virasoro minimal models $M(k,2k\pm1)$ by analysing the recently introduced half-lattice paths. These fermionic expressions display a quasiparticle formulation characteristic of the $\phi_{2,1}$ and $\phi_{1,5}$ integrable perturbations. We find that they arise by imposing a simple restriction on the RSOS quasiparticle states of the unitary models $M(p,p+1)$. In fact, four fermionic expressions are obtained for each generating function of half-lattice paths of finite length $L$, and these lead to four distinct expressions for most characters $\chi^{k,2k\pm1}_{r,s}$. These are direct analogues of Melzer's expressions for $M(p,p+1)$, and their proof entails revisiting, reworking and refining a proof of Melzer's identities which used combinatorial transforms on lattice paths. We also derive a bosonic version of the generating functions of length $L$ half-lattice paths, this expression being notable in that it involves $q$-trinomial coefficients. Taking the $L\to\infty$ limit shows that the generating functions for infinite length half-lattice paths are indeed the Virasoro characters $\chi^{k,2k\pm1}_{r,s}$.Comment: 29 pages. v2: minor improvements, references adde

Half-lattice paths and Virasoro characters

We first briefly review the role of lattice paths in the derivation of fermionic expressions for the M(p,p') minimal model characters of the Virasoro Lie algebra. We then focus on the recently introduced half-lattice paths for the M(p,2p+/-1) characters, reformulating them in such a way that the two cases may be treated uniformly. That the generating functions of these half-lattice paths are indeed M(p,2p+/-1) characters is proved by describing weight preserving bijections between them and the corresponding RSOS lattice paths. Here, the M(p,2p-1) case is derived for the first time. We then apply the methods of Bressoud and Warnaar to these half-lattice paths to derive fermionic expressions for the Virasoro characters X^{p,2p+/-1}_{1,2} that differ from those obtained from the RSOS paths. This work is an extension of that presented by the third author at the "7th International Conference on Lattice Path Combinatorics and Applications", Siena, Italy, July 2010.Comment: 22 page

Two-Rowed Hecke Algebra Representations at Roots of Unity

In this paper, we initiate a study into the explicit construction of irreducible representations of the Hecke algebra $H_n(q)$ of type $A_{n-1}$ in the non-generic case where $q$ is a root of unity. The approach is via the Specht modules of $H_n(q)$ which are irreducible in the generic case, and possess a natural basis indexed by Young tableaux. The general framework in which the irreducible non-generic $H_n(q)$-modules are to be constructed is set up and, in particular, the full set of modules corresponding to two-part partitions is described. Plentiful examples are given.Comment: LaTeX, 9 pages. Submitted for the Proceedings of the 4th International Colloquium ``Quantum Groups and Integrable Systems,'' Prague, 22-24 June 199

An intracellular motif of GLUT4 regulates fusion of GLUT4-containing vesicles

<p>Abstract</p> <p>Background</p> <p>Insulin stimulates glucose uptake by adipocytes through increasing translocation of the glucose transporter GLUT4 from an intracellular compartment to the plasma membrane. Fusion of GLUT4-containing vesicles at the cell surface is thought to involve phospholipase D activity, generating the signalling lipid phosphatidic acid, although the mechanism of action is not yet clear.</p> <p>Results</p> <p>Here we report the identification of a putative phosphatidic acid-binding motif in a GLUT4 intracellular loop. Mutation of this motif causes a decrease in the insulin-induced exposure of GLUT4 at the cell surface of 3T3-L1 adipocytes via an effect on vesicle fusion.</p> <p>Conclusion</p> <p>The potential phosphatidic acid-binding motif identified in this study is unique to GLUT4 among the sugar transporters, therefore this motif may provide a unique mechanism for regulating insulin-induced translocation by phospholipase D signalling.</p