### Abstract

Let F be a field, and α _{0},...,α _{k-1} be k distinct elements of F. Let λ =(λ _{1},..., λ _{k}) be a partition of n and V _{λ} be the set of all vectors v=(v _{1},...,v _{n}) F ^{n} such that|{j ∈[n] : v _{j}=α _{i}}|=λ _{i+1} for 0≦ i ≦ k-1. We describe the lexicographic standard monomials of the ideal of polynomials from F[x _{1},...,x _{n}] which vanish on the set V _{λ}. In the proof we give a new description of the orthogonal complement (S ^{λ)}) ^{⊥} (with respect to the James scalar product) of the Specht module S ^{λ}. As applications, a basis of (S ^{λ}) ^{⊥} is exhibited, and we obtain a combinatorial description of the Hilbert function of V _{λ..}. Our approach gives also the deglex standard monomials of V _{λ}, and hence provides a new proof of a result of A. M. Garsia and C. Procesi [10].

Original language | English |
---|---|

Pages (from-to) | 193-212 |

Number of pages | 20 |

Journal | Acta Mathematica Hungarica |

Volume | 111 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 1 2006 |

### Keywords

- Gröbner basis
- Hilbert function
- Specht module
- Standard monomial
- Tableau

### ASJC Scopus subject areas

- Mathematics(all)

## Fingerprint Dive into the research topics of 'Standard monomials for partitions'. Together they form a unique fingerprint.

## Cite this

*Acta Mathematica Hungarica*,

*111*(3), 193-212. https://doi.org/10.1007/s10474-006-0049-1