I tried explaining other binomial coefficient equations using representation theory and failed. But there's also a table of "off by one" equations, and I can explain one of those:
\[ \binom{32}{2} = 496 \]
while
\[ \binom{12}{4} + 1 \]
The Lie algebra 𝔰𝔬(12) has a 12-dimensional vector representation V and a 32-dimensional chiral spinor representation S⁺ There seems to be an isomorphism of representations
Λ²(S⁺) ≅ Λ⁴(V) ⊕ Λ⁰(V)
and this gives that equation.
In fact this is part of an infinite family starting with Paul Schwahn (npub1zz2…9mhx)'s equation, but they get more and more complicated and less exciting each time!
Oscar Cunningham (npub10mu…2h9c)
John Carlos Baez
npub1nf…3nqe4
2026-09-20 16:18:33 UTC
in reply to nevent1q…0v0u
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2026-09-20 16:18:33 UTCEvent JSON
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"content": "I tried explaining other binomial coefficient equations using representation theory and failed. But there's also a table of \"off by one\" equations, and I can explain one of those:\n\n\\[ \\binom{32}{2} = 496 \\] \n\nwhile\n\n\\[ \\binom{12}{4} + 1 \\]\n\nThe Lie algebra 𝔰𝔬(12) has a 12-dimensional vector representation V and a 32-dimensional chiral spinor representation S⁺ There seems to be an isomorphism of representations\n\nΛ²(S⁺) ≅ Λ⁴(V) ⊕ Λ⁰(V) \n\nand this gives that equation. \n\nIn fact this is part of an infinite family starting with nostr:npub1zz2mah7y6j4nld7dwedl7eh0yph4swkkv88spuvh0urq32xgz3ws459mhx's equation, but they get more and more complicated and less exciting each time!\n\nnostr:npub10mu87ru8hnpc7rfaequx5r7vtmrr94ujhqv9xsrjd4nxmmg5vufsqp2h9c",
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