Abstract
We introduce the difference operator for functions defined on strict partitions and prove a polynomiality property for a summation involving the bar length (hook length) and content statistics. As an application, several new hook-content formulas for strict partitions are derived.
Highlights
The basic knowledge on partitions, Young tableaux and symmetric functions could be found in [20]
A famous result in representation of finite groups, Nakayama Conjecture, says that two irreducible characters of Sn lie in the same p-block if and only if their corresponding partitions have the same p-core
The irreducible spin characters of the covering groups of the alternating group An and the symmetric group Sn are determined by strict partitions with size n
Summary
The basic knowledge on partitions, Young tableaux and symmetric functions could be found in [20]. For the (i, j)-box in the shifted Young diagram of the strict partition λ, we can associate its bar length (in some other papers, it is called hook length), denoted by h(i,j), which is the number of boxes. Consider the box 2 = (1, 3) in the shifted Young diagram of the strict partition (7, 5, 4, 1). Where Hν denotes the product of all hook lengths of boxes in ν and fν denotes the number of standard Young tableaux of shape ν. For two strict partitions λand μ, we write λ ⊇ μif λi ≥ μi for any i ≥ 1 In this case, the skew strict partition λ/μis identical with its skew shifted Young diagram.
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