SHc^c Realization of Minimal Model CFT: Triality, Poset and Burge Condition

Recently an orthogonal basis of $\mathcal{W}_N$-algebra (AFLT basis) labeled by $N$-tuple Young diagrams was found in the context of 4D/2D duality. Recursion relations among the basis are summarized in the form of an algebra SH$^c$ which is universal for any $N$. We show that it has an $\mathfrak{S}_3$ automorphism which is referred to as triality. We study the level-rank duality between minimal models, which is a special example of the automorphism. It is shown that the nonvanishing states in both systems are described by $N$ or $M$ Young diagrams with the rows of boxes appropriately shuffled. The reshuffling of rows implies there exists partial ordering of the set which labels them. For the simplest example, one can compute the partition functions for the partially ordered set (poset) explicitly, which reproduces the Rogers-Ramanujan identities. We also study the description of minimal models by SH$^c$. Simple analysis reproduces some known properties of minimal models, the structure of singular vectors and the $N$-Burge condition in the Hilbert space.Comment: 1+38 pages and 12 figures. v2: typos corrected + comments adde

Distributed Environment Control Using Wireless Sensor/Actuator Networks for Lighting Applications

We propose a decentralized algorithm to calculate the control signals for lights in wireless sensor/actuator networks. This algorithm uses an appropriate step size in the iterative process used for quickly computing the control signals. We demonstrate the accuracy and efficiency of this approach compared with the penalty method by using Mote-based mesh sensor networks. The estimation error of the new approach is one-eighth as large as that of the penalty method with one-fifth of its computation time. In addition, we describe our sensor/actuator node for distributed lighting control based on the decentralized algorithm and demonstrate its practical efficacy