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Properties of Four Partial Orders on Standard Young Tableaux

Let SY Tn be the set of all standard Young tableaux with n cells. After recalling the definitions of four partial orders, the weak, KL, geometric and chain orders on SY Tn and some of their crucial properties, we prove three main results: • Intervals in any of these four orders essentially describe the product in a Hopf algebra of tableaux defined by Poirier and Reutenauer. • The map sending a tableau to its descent set induces a homotopy equivalence of the proper parts of all of these orders on tableaux with that of the Boolean algebra 2 [n−1]. In particular, the Möbius function of these orders on tableaux is (−1) n−3. • For two of the four orders, one can define a more general order on skew tableaux having fixed inner boundary, and similarly analyze their homotopy type and Möbius function.