Contributions à l'Etude du Vertex Topologique en Théorie ... - Toubkal

013509-10 Drissi, Jehjouh, and Saidi J. Math. Phys. 50, 013509 2009FIG. 1. Shifted Young diagram of =4,2,1.bThe topological vertex T ˆ ˆ ˆq,t describing the refined version shifted topological vertexS ˆ ˆ ˆq. Putting ˆ =ˆ =ˆ =, wegetT 3 q,t = j=1 □i=1 i1+qj−1 t i1−q j−1 t5.1describing the refined version of Foda–Wheeler relation recovered by setting t=q.In the end, notice that it would be interesting to seek whether T ˆ ˆ ˆq,t could be associatedwith some gauge theory instantons as does R q,t with the Nekrasov’s ones.ACKNOWLEDGMENTSThis research work was supported by Protars III D12/25.APPENDIX: STRICT PARTITION AND SCHUR FUNCTIONIn this Appendix, we give some useful tools on the strict 2d-partitiions, the shifted planepartitions and on Schur functions.1. Strict 2d- and shifted 3d-partitionA2d-partition, or a Young diagram, denoted as = 1 , 2 ,..., r ,... is a sequence of decreasingnon-negative integers 1 2¯ r ¯.A strict 2d-partition is a sequence of strictly decreasing integers 1 2 ¯. The sum of theparts i of the 2d-partition is the weight of denoted by = 1 + 2 + ¯ + r + ¯ .A2d strict partition is said a partition of n if =n and is represented by its shifted Youngdiagram obtained from the usual diagram by shifting to the right the ith row by i−1 squares asshown on Fig. 1.The shifted Young diagram is given by a collection of boxes with coordinatesA1i, ji =1, ...,l, i j i + i −1. A2A shifted plane partition of shape is determined by the sequence ...,. −1 , 0 , 1 ,..., where 0 the 2d-partition on the main diagonal and k is the 2d-partition on the diagonal shifted by aninteger k. All diagonal partitions are strict 2d-partitions forming altogether a shifted plane partitionwith the propertyDownloaded 24 Mar 2009 to 140.105.16.64. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp