Skew tableau

Definition

A skew tableau is a graphical representation used in combinatorial representation theory, formed by placing numbers in a Young diagram that is shaped like a skew shape, which is created by removing a smaller rectangular Young diagram from a larger one. This structure helps in studying the representations of symmetric groups and allows for the visualization of various combinatorial objects and their properties, particularly in the context of decomposing tensor products of representations.

5 Must Know Facts For Your Next Test

1. Skew tableaux are particularly useful in the context of representing the decomposition of tensor products into irreducible components.
2. Each skew tableau corresponds to a unique way of labeling the boxes that maintains certain rules about the arrangement of the numbers.
3. The hook-length formula can be applied to skew tableaux to count the number of standard fillings, providing insights into their combinatorial structure.
4. Skew tableaux can be used to represent particular basis elements in the space of representations, illustrating their relevance to character theory.
5. The study of skew tableaux connects to various areas such as Schur functions and Macdonald polynomials, enriching the understanding of symmetric functions.

Review Questions

• How do skew tableaux contribute to understanding the decomposition of tensor products in representation theory?
  • Skew tableaux play a significant role in visualizing how tensor products decompose into irreducible representations. They provide a way to organize and count the different ways that representations can combine, facilitating the application of tools like the hook-length formula. By analyzing the structure of skew tableaux, one can determine how many irreducible components arise from a given tensor product.
• What are the conditions that must be met when filling out a skew tableau with numbers, and why are these conditions important?
  • When filling out a skew tableau, the numbers must increase across each row and down each column, similar to regular Young tableaux. These conditions ensure that the filling respects the combinatorial structure of the tableau and maintains order, which is critical for establishing relationships between different representations. Adhering to these rules ensures that each tableau corresponds accurately to an element in representation theory.
• Evaluate the significance of skew tableaux in relation to Schur functions and their applications in representation theory.
  • Skew tableaux are deeply connected to Schur functions, which are generating functions that encode information about symmetric polynomials. The significance lies in how skew tableaux correspond to specific Schur functions, facilitating calculations related to character theory and representation decomposition. By evaluating these connections, one can gain insights into how representations behave under various operations and how they relate to combinatorial identities.

Related terms

Young tableau: A Young tableau is a way of filling the boxes of a Young diagram with numbers that increase across each row and down each column, often used to study representations of symmetric groups.

Symmetric group: The symmetric group is the group of all permutations of a finite set, playing a crucial role in representation theory and combinatorial structures.

Tensor product: The tensor product is an operation on two representations that combines them into a new representation, allowing for the analysis of more complex representations through simpler ones.