Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: The weighted hook length formula III: Shifted tableaux. | The weighted hook length formula III Shifted tableaux Matjaz Konvalinka Submitted Jan 12 2011 Accepted Apr 22 2011 Published May 8 2011 Mathematics Subject Classification 05A17 05A19 05E10 Abstract Recently a simple proof of the hook length formula was given via the branching rule. In this paper we extend the results to shifted tableaux. We give a bijective proof of the branching rule for the hook lengths for shifted tableaux present variants of this rule including weighted versions and make the first tentative steps toward a bijective proof of the hook length formula for d-complete posets. 1 Introduction Let A A1 A2 . A Al A2 . A 0 be a partition of n A L n and let A i j G Z2 1 i k 1 j Ai be the corresponding Young diagram. The conjugate partition A is defined by Aj max i Ai j . The hook Hz Hij c A is the set of squares weakly to the right and below of z i j G A and the hook length hz hij Hz Ai Aj i j 1 is the size of the hook. The punctured hook Hz c A is the set Hz z . See Figure 1 left and center drawing. We write n for 1 . n P A for the power set of A and A u B for the disjoint union of not necessarily disjoint sets A and B. Furthermore for squares i j and i j of A write i j i j if i Í and j j . A standard Young tableau of shape A is a bijective map f A n such that f z f z whenever z z and z z . See Figure 1 right drawing. We denote the number of standard Young tableaux of shape A by f . The remarkable hook length formula states that if A is a partition of n then f _ n Hze A hz Department of Mathematics University of Ljubljana and Institute for Mathematics Physics and Mechanics Ljubljana THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P101 1 For example for A 3 2 2 F 7 the hook length formula gives 7 322 21. J 5 4 1 3 2 2 1 1 2 7 3 5 4 6 Figure 1 Young diagram A A 66532 hook H23 with hook length h23 6 left punctured hook H31 center a standard Young tableau of shape 322 right . This gives a short formula for dimensions of irreducible
