Semi-standard young tableau

Definition

A semi-standard young tableau is a way to fill a Young diagram with positive integers that weakly increase across each row and strictly increase down each column. This structure is crucial in combinatorics, particularly in representing the RSK correspondence, where each tableau corresponds to a specific permutation and provides insights into the representation theory of symmetric groups.

5 Must Know Facts For Your Next Test

1. In a semi-standard young tableau, the entries are filled with positive integers such that numbers in each row are non-decreasing and numbers in each column are strictly increasing.
2. The shape of a semi-standard young tableau corresponds to a partition, which dictates how many boxes are placed in each row.
3. The RSK correspondence transforms a permutation into a pair of semi-standard young tableaux, which reflects important combinatorial properties of that permutation.
4. The concept of semi-standard young tableaux is extensively used in algebraic geometry and representation theory, particularly in the study of symmetric functions.
5. Each semi-standard young tableau can be associated with a weight vector, where the weight corresponds to the sum of the entries within the tableau.

Review Questions

• How does the structure of a semi-standard young tableau facilitate the understanding of permutations through the RSK correspondence?
  • The structure of a semi-standard young tableau allows for a clear representation of permutations by ensuring that each tableau reflects the original ordering and relationships present in the permutation. Under the RSK correspondence, each permutation can be transformed into two tableaux: one representing the insertion process and another capturing the output. This dual representation enables combinatorialists to study permutations in an organized manner, revealing underlying patterns and properties.
• Discuss the importance of dominance order in relation to semi-standard young tableaux and how it influences their properties.
  • Dominance order plays a significant role in understanding the relationships between different semi-standard young tableaux. When comparing tableaux shapes, dominance order can determine which tableaux can be obtained from others through specific operations or transformations. It helps characterize how these tableaux interact under various combinatorial operations, influencing their behavior in representations and contributing to the rich structure present in combinatorial theory.
• Evaluate how semi-standard young tableaux contribute to advancements in representation theory and their applications beyond basic combinatorics.
  • Semi-standard young tableaux significantly enhance representation theory by providing tools for constructing representations of symmetric groups and studying their character theory. Their structure leads to new insights into algebraic geometry, particularly through geometric interpretation via Schubert varieties. As these tableaux connect various areas such as algebraic topology and mathematical physics, they foster interdisciplinary collaborations and inspire further research into complex mathematical problems.

Related terms

Young diagram: A graphical representation of partitions, consisting of rows of boxes aligned to the left, where the number of boxes in each row corresponds to the parts of the partition.

Robinson-Schensted-Knuth (RSK) correspondence: A combinatorial algorithm that establishes a connection between permutations and pairs of semi-standard young tableaux, providing a systematic way to study their properties.

Dominance order: A relation on partitions where one partition is said to dominate another if, when comparing their parts cumulatively, it always has at least as many parts that are greater than or equal.