OAI.RowColumn.occupied_overlap

The theorem states that there is a constant C>0 such that the following holds for all board dimensions m and n, every set Ω of occupied cells of the m-by-n board Fin m × Fin n, every Young diagram a with a bijection e from Ω to the cells of a (so |a|=|Ω|), and every integer h with 1≤h≤mn such that a lies in the (h,h)-hook, meaning every zero-based cell (i,j) of a has i<h or j<h. Let the row group be the permutations of Ω preserving each cell's row, and the column group be those preserving each cell's column. Let the Specht module of a be the span, inside complex functions on tabloids of a (viewed as a Hilbert space), of the permutation-translates of the polytabloid, which is the column-group sign-alternated image of the base tabloid. Relabelling by e makes it a unitary representation of the permutations of Ω, restricted to the row group and to the column group. Take any irreducible unitary representation ρ of the row group on ℂ^r and any irreducible unitary representation τ of the column group on ℂ^c. Take any families I of u and J of v linear isometries from ℂ^r and ℂ^c, respectively, into the Specht module, each family being complete copies: each map intertwines the group actions, different members have mutually orthogonal images, and every intertwiner has range contained in the sum of the members' ranges, so the family exhausts the isotypic component. Then the sum over all i and j of the squared operator norm of I_i^*J_j is at most exp(C((m+n+1)h² log(mn+2) + (mn−|Ω|))) · min(1, rc/dim of the Specht module).