next | previous | forward | backward | up | index | toc

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

## Synopsis

• Usage:
numDistinctRepsBetti(d,n,b)
• Inputs:
• Outputs:
• ,

## 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 :

• numDistinctRepsBetti(ZZ,ZZ,ZZ)

## For the programmer

The object numDistinctRepsBetti is .