Are there 10-dimensional irreducible representations of the Lie algebra $so(10,\mathbb{C})$ which are not isomorphic to the standard representation?
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3$\begingroup$ You want “irreducible”? (Otherwise take enough copies of the trivial rep…) $\endgroup$– Sam HopkinsCommented Apr 4, 2022 at 17:48
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1$\begingroup$ Yes. Corrected. Thank you. $\endgroup$– asvCommented Apr 4, 2022 at 18:09
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1 Answer
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A simple application of the Weyl degree formula shows that the degrees of the fundamental representations of $D_5$ are $10,45,120,16,16$ (the $10$-dimensional fundamental representation being the standard one, the $45$-dimensional being the adjoint one, and the two $16$-dimensional ones being the two half-spin representations). Computation shown here in Sage:
sage: D5 = WeylCharacterRing("D5")
sage: [D5(D5.fundamental_weights()[i]).degree() for i in range(1,6)]
[10, 45, 120, 16, 16]
Since the degree of an irreducible representation can only increase when we increase the highest weight (by adding fundamental weights), the only $10$-dimensional representations of $D_5$ are the standard one and a sum of $10$ copies of the trivial representation.
