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numDistinctRepsBetti -- a hash table containing the number of distinct Schur functors of a Veronese embedding

Synopsis

Description

This function returns a hash table $H$ whose keys are pairs $(p,q)$ such that the corresponding value $H#(p,q)$ is the number of distinct Schur functors appearing in the Schur functor decomposition of $K_{p,q}(\mathbb{P}^n, d;b)$. Note that the function numRepsBetti is similar, though it counts Schur functors appearing with multiplicity.

i1 : totalBettiTally(3,2,0)

            0  1   2   3   4   5  6 7
o1 = total: 1 27 105 189 189 105 27 1
         0: 1  .   .   .   .   .  . .
         1: . 27 105 189 189 105 27 .
         2: .  .   .   .   .   .  . 1

o1 : BettiTally
 
i2 : numDistinctRepsBetti(3,2,0)

o2 = HashTable{(0, 0) => 1}
               (0, 1) => 0
               (0, 2) => 0
               (1, 0) => 0
               (1, 1) => 1
               (1, 2) => 0
               (2, 0) => 0
               (2, 1) => 4
               (2, 2) => 0
               (3, 0) => 0
               (3, 1) => 7
               (3, 2) => 0
               (4, 0) => 0
               (4, 1) => 7
               (4, 2) => 0
               (5, 0) => 0
               (5, 1) => 4
               (5, 2) => 0
               (6, 0) => 0
               (6, 1) => 1
               (6, 2) => 0
               (7, 0) => 0
               (7, 1) => 0
               (7, 2) => 1

o2 : HashTable
 
i3 : makeBettiTally oo

            0 1 2 3 4 5 6 7
o3 = total: 1 1 4 7 7 4 1 1
         0: 1 . . . . . . .
         1: . 1 4 7 7 4 1 .
         2: . . . . . . . 1

o3 : BettiTally

This example shows that the $\beta_{3,4}(3,2,0)$ entry, which is $189$, consists of $7$ distinct Schur functors. The specific Schur functors that appear can be computed using schurBetti. Here is a more complicated example:

i4 : totalBettiTally(5,2,1)

            0  1   2    3     4     5      6      7      8      9     10     11     12    13    14   15  16 17 18
o4 = total: 3 35 525 5865 29988 97920 231540 417690 590070 661232 590070 417690 231540 97920 29988 5865 525 35  3
         0: 3 35 120    .     .     .      .      .      .      .      .      .      .     .     .    .   .  .  .
         1: .  . 405 5865 29988 97920 231540 417690 590070 661232 590070 417690 231540 97920 29988 5865 405  .  .
         2: .  .   .    .     .     .      .      .      .      .      .      .      .     .     .    . 120 35  3

o4 : BettiTally
 
i5 : makeBettiTally numDistinctRepsBetti(5,2,1)

            0 1 2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18
o5 = total: 1 1 5 26 43 53 63 68 72 73 72 68 63 53 43 26  5  1  1
         0: 1 1 3  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
         1: . . 2 26 43 53 63 68 72 73 72 68 63 53 43 26  2  .  .
         2: . . .  .  .  .  .  .  .  .  .  .  .  .  .  .  3  1  1

o5 : BettiTally

Ways to use numDistinctRepsBetti :

For the programmer

The object numDistinctRepsBetti is a method function.