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A Young tableau is obtained by filling in the boxes of the Young diagram with symbols taken from some alphabet, which is usually required to be a totally ordered set. Originally that alphabet was a set of indexed variables x 1, x 2, x 3..., but now one usually uses a set of numbers for brevity.
There are various ways to associate a word (in the sense of combinatorics, i. e., a finite sequence of elements of an alphabet—here the set of positive integers) to every Young tableau. We choose the one apparently most popular: We associate to every Young tableau T the word obtained by concatenating the rows of T from the bottom row to the ...
• The product then has the tableau form (8)(35)(1247) Every word is Knuth equivalent to the word of a unique semistandard Young tableau (this means that each row is non-decreasing and each column is strictly increasing) over the same ordered alphabet, where the tableau may be read by rows or by columns. So the elements of the plactic monoid ...
A Young tableau (pl.: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups and to study their properties. Young tableaux were introduced by Alfred Young, a mathematician at Cambridge University, in 1900
Let = be a partition of = + +.It is customary to interpret graphically as a Young diagram, namely a left-justified array of square cells with rows of lengths , …,.A (standard) Young tableau of shape is a filling of the cells of the Young diagram with all the integers {, …,}, with no repetition, such that each row and each column form increasing sequences.
For any partition , the Kostka number is equal to 1: the unique way to fill the Young diagram of shape = (, …,) with copies of 1, copies of 2, and so on, so that the resulting tableau is weakly increasing along rows and strictly increasing along columns is if all the 1s are placed in the first row, all the 2s are placed in the second row, and so on.
Theo Wargo/Getty. Tyrus and his wife Ingrid Rinck with their kids attends FOX Nation's 2024 Patriot Awards at Tilles Center for the Performing Arts on December 05, 2024 in Greenvale, New York.
Schur functors are indexed by partitions and are described as follows. Let R be a commutative ring, E an R-module and λ a partition of a positive integer n.Let T be a Young tableau of shape λ, thus indexing the factors of the n-fold direct product, E × E × ... × E, with the boxes of T.