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

Synopsis

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.