How many standard Young tableaux exist for the partition (3, 2) of 5?
A3
B5
C7
D10
Using the hook length formula: dim = 5! / (h₁₁·h₁₂·h₁₃·h₂₁·h₂₂). The hook lengths for the (3,2) diagram are: h₁₁=4, h₁₂=3, h₁₃=1, h₂₁=2, h₂₂=1. So dim = 120/(4·3·1·2·1) = 120/24 = 5. One can verify by listing: the 5 standard fillings of a 3-box row atop a 2-box row with {1,2,3,4,5} increasing along rows and down columns.
Question 2 True / False
A standard Young tableau must have entries increasing along rows (left to right) and down columns (top to bottom). If a filling violates the column condition but satisfies the row condition, is it still a standard tableau?
TTrue
FFalse
Answer: False
A standard Young tableau requires BOTH conditions: strict increase along rows and strict increase down columns. A filling satisfying only the row condition is not standard. The column condition is essential — it ensures that the corresponding element in the group algebra generates an irreducible submodule. Dropping either condition changes the combinatorial count and breaks the connection to representation theory.
Question 3 Multiple Choice
The hook length of a box in position (i, j) of a Young diagram counts:
AThe number of boxes directly below the box
BThe number of boxes directly to its right
CThe number of boxes directly to its right, directly below it, plus the box itself
DThe total number of boxes in row i plus column j
The hook of box (i,j) consists of the box itself, all boxes directly to its right in the same row, and all boxes directly below it in the same column. The hook length h(i,j) is the count of these boxes. For the partition (3,2), the box at position (1,1) has hook {(1,1),(1,2),(1,3),(2,1)} — 2 boxes to the right, 1 box below, plus itself — giving h(1,1) = 4.
Question 4 Short Answer
Why do Young diagrams appear in the representation theory of both Sₙ and GL_n(ℂ)?
Think about your answer, then reveal below.
Model answer: Schur-Weyl duality establishes that the actions of Sₙ and GL_n(ℂ) on tensor space (ℂⁿ)^{⊗n} are mutual centralizers. Decomposing this tensor space under both actions simultaneously produces a correspondence: irreducible representations of Sₙ (indexed by partitions of n) pair with irreducible polynomial representations of GL_n (indexed by the same partitions). Young diagrams serve as the common indexing set for both.
This is one of the deepest connections in representation theory. The partition λ labels a Specht module for Sₙ and a Schur functor for GL_n. The semistandard Young tableaux of shape λ with entries in {1,...,n} index a basis for the GL_n-representation, while the standard tableaux of shape λ index a basis for the Sₙ-representation. The combinatorics of tableaux thus serves as a bridge between two seemingly different representation theories.