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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6320 • Algebra II

Summer 2021 • Question 3

Question

Math 6320 • Algebra II Summer 2021 • Question 3

Recall that the standard representation of \(S_n\) is the subspace

\[ V_n := \{(x_1,\ldots ,x_n)\in \mathbb {C}^n,\ x_1+x_2+\cdots +x_n=0\} \]

with \(S_n\) acting by permutation of the coordinates. Show that \(\wedge ^2 V_5\) is an irreducible representation of \(S_5\).

Raw LaTeX

Recall that the standard representation of \(S_n\) is the subspace

\[
V_n := \{(x_1,\ldots,x_n)\in\mathbb{C}^n,\ x_1+x_2+\cdots+x_n=0\}
\]


with \(S_n\) acting by permutation of the coordinates. Show that
\(\wedge^2 V_5\) is an irreducible representation of \(S_5\).

Topic Tags

Irreducible Polynomials Representation Theory Symmetric and Alternating Groups