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Identifying Young diagrams among residue multisets

Salim Rostam

Abstract

To any Young diagram we can associate the multiset of residues of all its nodes. This paper is concerned with the inverse problem: given a multiset of elements of Z/eZ, does it comes from a Young diagram? We give a full solution in level one and a partial answer in higher levels for Young multidiagrams, using Fayers's notions of core block and weight of a multipartition. We apply the result in level one to study a shift operation on partitions.

Identifying Young diagrams among residue multisets

Abstract

To any Young diagram we can associate the multiset of residues of all its nodes. This paper is concerned with the inverse problem: given a multiset of elements of Z/eZ, does it comes from a Young diagram? We give a full solution in level one and a partial answer in higher levels for Young multidiagrams, using Fayers's notions of core block and weight of a multipartition. We apply the result in level one to study a shift operation on partitions.

Paper Structure

This paper contains 22 sections, 81 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Background on core blocks
  3. Partitions
  4. Abaci
  5. Blocks for partitions
  6. Multipartitions
  7. Blocks for multipartitions
  8. Implicit description of the blocks
  9. Weight of a tuple of subsets
  10. Bound for the weight of core blocks
  11. Optimality
  12. Values for small parameters
  13. Bounds
  14. More values for small parameters
  15. With small r
  16. ...and 7 more sections

Key Result

Theorem A

Let $r \geq 2$ and $e > 0$. We have where $N'_{r,e} \coloneqq \left\lfloor\frac{r^2}{2e}\left\lfloor\frac{e^2}{4}\right\rfloor\right\rfloor$.

Theorems & Definitions (172)

  • Theorem A: Corollaries \ref{['corollary:Qmcharge_supset_superlevel_weight']} and \ref{['corollary:sharpest_upper_bound_bestboundcoreblock']}
  • Theorem B: Propositions \ref{['proposition:partitions_wgeq0']} and \ref{['proposition:Qmcharge_semialg_particular_case']}
  • Theorem C: Corollary \ref{['corollary:case_spart_lambda_good']}
  • Theorem D: Lemma \ref{['lemma:stuttering_partition']} and Corollary \ref{['corollary:stuttering_blocks']}
  • Example 2.1
  • Definition 2.2
  • Example 2.3
  • Remark 2.5
  • Lemma 2.6
  • Lemma 2.8
  • ...and 162 more