OAI.RowColumn.occupied_overlap_endpoint

The theorem states that there is a constant C>0 such that the following bound holds for all board sizes m,n, every finite set Ω of cells of the m×n board, every Young diagram a with a bijection e from Ω to the cells of a (so |a|=|Ω|), and every h with 1≤h≤mn such that a lies in the (h,h)-hook, meaning every cell (i,j) of a, zero-based, has i<h or j<h. Let the row group be the permutations of Ω preserving each cell's row index, and the column group those preserving each column index. The Specht module of a is the span of the permutation-orbit of the polytabloid (the column-sign alternator applied to the base tabloid) inside the complex functions on tabloids, viewed as a Hilbert space with a unitary action, and e relabels it as a representation of permutations of Ω, restricted to the row and column groups. Take irreducible unitary representations ρ of the row group on ℂ^r and τ of the column group on ℂ^c, and families I₁,…,I_u and J₁,…,J_v of linear isometries from ℂ^r and ℂ^c into this Specht Hilbert space, where the I family is a complete set of copies of ρ and the J family a complete set of copies of τ. Complete copies means that each map intertwines the given representation with the restricted Specht action, the maps have mutually orthogonal ranges, and the range of every intertwiner from ℂ^r (respectively ℂ^c) into the Specht space lies in the sum of the ranges of the family. Then the sum over all i and j of the squared operator norms ‖Iᵢ*Jⱼ‖² is at most exp(C·((m+n+1)·h²·log(mn+2) + (mn−|Ω|))) · min(1, rc/dim of the Specht module), where mn−|Ω| is a natural-number subtraction.