Common use of Combinatorics of partitions Clause in Contracts

Combinatorics of partitions. A partition λ of length n is an n-tuple of integers (l1, l2, . . . , ln) such that l1 ≥ l2 ≥ · · · ≥ ln. In this thesis we will allow partitions to have negative entries. For example µ := (5, 3, 3, 1, −1) is a partition of length 5. For a partition λ = (l1, l2, . . . , ln) define Σ| | λ = li. For our example µ we have |µ| = 11. We can associate to a partition λ = (l1, l2, . . . , ln) a diagram called a Young diagram as follows. Draw a line L and split it into n equal rows labelled 1, 2, . . . , n from top to bottom. For 1 ≤ i ≤ n if li ≥ 0 add li boxes in row i to the right of the line and if li < 0 add |li| boxes to the left of the line. For our example µ, the corresponding Young diagram is displayed in Figure 2.1 where we have coloured boxes green to show they are to the left of the line L and correspond to negative parts in µ. Figure 2.1: The Young diagram corresponding to (5, 3, 3, 1, −1) For a prime p and a pair of integers (a, b) define the p-residue, res(a, b) to be the element of Z /p Z congruent to (b−a) mod p. For an element α ∈ Z /p Z and a partition λ = (l1, l2, . . . , ln) we define the integer contα(λ) :=|{(a, b)|1 ≤ a ≤ n, 0 < b ≤ la, res(a, b) = α}| − |{(a, b)|1 ≤ a ≤ n, la < b ≤ 0, res(a, b) = α}|. One way to understand contα(λ) is to label the corresponding Young diagram with p-residues in the following way. For 1 ≤ i ≤ n if li ≥ 0 then label the jth box from the left in row i of the Young diagram with the value j − i mod p. If li < 0 then label the jth box from the right in row i of the Young diagram with the value −j + 1 − i mod p. The following diagram shows the Young diagram of µ labelled with 5-residues. 0 The value contα(λ) is then the number of times α ∈ Z /p Z appears in the p-residue labelled Young diagram to the right of the line L minus the number of times it appears to the left. So for example cont0(µ) = 3 − 1 = 2 and cont4(µ) = 3. 2.2.1. We say two partitions λ and µ are p-linked and write λ ∼p µ if for all α ∈ Z /p Z we have contα(λ) = contα(µ). In particular we write λ p µ if there exists some α ∈ Z /p Z such that contα(λ) /= contα(µ). When we use linkage we will do so in a context where p is a fixed prime which is the characteristic of the field F and the defining characteristic of the group GLn(F). In this context we will drop the subscript p and just write λ ∼ µ.