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numRepsBetti -- a hash table containing the number of 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 Schur functors appearing in the Schur functor decomposition of $K_{p,q}(\mathbb{P}^n, d;b)$ counted with multiplicity. Note that the function numDistinctRepsBetti is similar, though that ignores the multiplicities of the Schur functors.

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 : numRepsBetti(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 $7$ Schur functors. (In this case, the Schur functors all appear with multiplicity 1.) The specific Schur functors that appear can be computed using schurBetti. Here is an example where some Schur functors appear with higher multiplicity.

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 numRepsBetti(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 38 165 477 1027 1733 2355 2608 2355 1733 1027 477 165 38  5  1  1
         0: 1 1 3  .   .   .    .    .    .    .    .    .    .   .   .  .  .  .  .
         1: . . 2 38 165 477 1027 1733 2355 2608 2355 1733 1027 477 165 38  2  .  .
         2: . . .  .   .   .    .    .    .    .    .    .    .   .   .  .  3  1  1

o5 : BettiTally

Ways to use numRepsBetti :

For the programmer

The object numRepsBetti is a method function.