Combinatorics of partitions Sample Clauses

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.
View SourceView Similar (2)

Related to Combinatorics of partitions

  • Combination and Split Up The Depositary, subject to the terms and conditions of this Deposit Agreement shall, upon surrender of a Receipt or Receipts for the purpose of effecting a split-up or combination of such Receipt or Receipts and upon payment to the Depositary of the applicable fees and charges set forth in Section 5.9 hereof and Article (9) of the Receipt, execute and Deliver a new Receipt or Receipts for any authorized number of American Depositary Shares requested, evidencing the same aggregate number of American Depositary Shares as the Receipt or Receipts surrendered.

  • Combinations and Split-ups of Receipts Upon surrender of a Receipt or Receipts at the Depositary Office or such other office as the Depositary may designate for the purpose of effecting a split-up or combination of Receipts, subject to the terms and conditions of this Deposit Agreement, the Depositary shall execute and deliver a new Receipt or Receipts in the authorized denominations requested evidencing the same aggregate number of Depositary Shares evidenced by the Receipt or Receipts surrendered.

  • Stock Splits, Subdivisions, Reclassifications or Combinations If the Company shall (i) declare and pay a dividend or make a distribution on its Common Stock in shares of Common Stock, (ii) subdivide or reclassify the outstanding shares of Common Stock into a greater number of shares, or (iii) combine or reclassify the outstanding shares of Common Stock into a smaller number of shares, the number of Shares issuable upon exercise of this Warrant at the time of the record date for such dividend or distribution or the effective date of such subdivision, combination or reclassification shall be proportionately adjusted so that the Warrantholder after such date shall be entitled to purchase the number of shares of Common Stock which such holder would have owned or been entitled to receive in respect of the shares of Common Stock subject to this Warrant after such date had this Warrant been exercised immediately prior to such date. In such event, the Exercise Price in effect at the time of the record date for such dividend or distribution or the effective date of such subdivision, combination or reclassification shall be adjusted to the number obtained by dividing (x) the product of (1) the number of Shares issuable upon the exercise of this Warrant before such adjustment and (2) the Exercise Price in effect immediately prior to the record or effective date, as the case may be, for the dividend, distribution, subdivision, combination or reclassification giving rise to this adjustment by (y) the new number of Shares issuable upon exercise of the Warrant determined pursuant to the immediately preceding sentence.

  • Stock Splits and Combinations If the Company shall at any time subdivide or combine its outstanding shares of Common Stock, this Warrant shall, after that subdivision or combination, evidence the right to purchase the number of shares of Common Stock that would have been issuable as a result of that change with respect to the shares of Common Stock which were purchasable under this Warrant immediately before that subdivision or combination. If the Company shall at any time subdivide the outstanding shares of Common Stock, the Warrant Exercise Price then in effect immediately before that subdivision shall be proportionately decreased, and, if the Company shall at any time combine the outstanding shares of Common Stock, the Warrant Exercise Price then in effect immediately before that combination shall be proportionately increased. Any adjustment under this section shall become effective at the close of business on the date the subdivision or combination becomes effective.

  • Recapitalizations, etc The provisions of this Agreement shall apply, to the full extent set forth herein with respect to the Stock or the Options, to any and all shares of capital stock of the Company or any capital stock, partnership units or any other security evidencing ownership interests in any successor or assign of the Company (whether by merger, consolidation, sale of assets or otherwise) which may be issued in respect of, in exchange for, or substitution of the Stock or the Options by reason of any stock dividend, split, reverse split, combination, recapitalization, liquidation, reclassification, merger, consolidation or otherwise.