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

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 .