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# schurBetti -- a hash table for Schur module decomposition of Veronese Betti tables

- Usage:
    `schurBetti(d,n,b)`
- Inputs:
  - d, an integer,
  - n, an integer,
  - b, an integer,
- Outputs:
  - a hash table,

## Description

This function returns a hash table with the Schur functor decompositions of the syzygies of $\mathcal{O}(b)$ under the embedding by $\mathcal{O}(d)$. The keys of the hash table $H$ are pairs $(p,q)$ where $H#(p,q)$ gives the Schur functor decomposition of $K_{p,q}(\mathbb{P}^n, d;b)$. $\mathcal{O}(b)$ the Schur functor decomposition as a list of tuples $(\{a_1,a_2,a_3\},b)$ where $\{a_1,a_2,a_3\}$ specifies the weight of the Schur functor and $m$ the multiplicity with which that particular Schur functor appears in the decomposition of $K_{p,q}(\mathbb{P}^n, d;b)$.

Some tables are incomplete and we mark unknown entries with ({0,0,0},infinity).

    i1 : schurBetti(3,2,0)

    o1 = HashTable{(0, 0) => {({0, 0, 0}, 1)}                                                                                                }
                   (0, 1) => {}
                   (0, 2) => {}
                   (1, 0) => {}
                   (1, 1) => {({4, 2, 0}, 1)}
                   (1, 2) => {}
                   (2, 0) => {}
                   (2, 1) => {({6, 2, 1}, 1), ({5, 4, 0}, 1), ({5, 3, 1}, 1), ({4, 3, 2}, 1)}
                   (2, 2) => {}
                   (3, 0) => {}
                   (3, 1) => {({7, 4, 1}, 1), ({7, 3, 2}, 1), ({6, 5, 1}, 1), ({6, 4, 2}, 1), ({6, 3, 3}, 1), ({5, 5, 2}, 1), ({5, 4, 3}, 1)}
                   (3, 2) => {}
                   (4, 0) => {}
                   (4, 1) => {({8, 5, 2}, 1), ({8, 4, 3}, 1), ({7, 6, 2}, 1), ({7, 5, 3}, 1), ({7, 4, 4}, 1), ({6, 6, 3}, 1), ({6, 5, 4}, 1)}
                   (4, 2) => {}
                   (5, 0) => {}
                   (5, 1) => {({9, 5, 4}, 1), ({8, 7, 3}, 1), ({8, 6, 4}, 1), ({7, 6, 5}, 1)}
                   (5, 2) => {}
                   (6, 0) => {}
                   (6, 1) => {({9, 7, 5}, 1)}
                   (6, 2) => {}
                   (7, 0) => {}
                   (7, 1) => {}
                   (7, 2) => {({9, 9, 9}, 1)}

    o1 : HashTable

## Ways to use schurBetti:

- `schurBetti(ZZ,ZZ,ZZ)`

## For the programmer

The object schurBetti is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/SchurVeronese/doc.m2:180:0.