{"id":8351,"date":"2026-09-21T18:44:21","date_gmt":"2026-09-22T01:44:21","guid":{"rendered":"https:\/\/theoryofeverything.org\/theToE\/?p=8351"},"modified":"2026-09-21T18:44:24","modified_gmt":"2026-09-22T01:44:24","slug":"group-theoretic-binomial-collision-analysis","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2026\/09\/21\/group-theoretic-binomial-collision-analysis\/","title":{"rendered":"Group Theoretic Binomial Collision Analysis"},"content":{"rendered":"\n

Please refer to the Baez blog post here<\/a>.<\/p>\n\n\n

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Baez’s comments regarding G2 and E6 binomial collisions<\/figcaption><\/figure>\n<\/div>\n\n\n

My first reaction is to question whether the “isomorphism of the irreducible representations” is or is not over-constraining the binomial’s numerical pattern match.<\/p>\n\n\n\n

I found a possible alternate path via \u039b5<\/mn>5<\/annotation><\/semantics><\/math><\/sup> SU(4) = \u039b5<\/mn>5<\/annotation><\/semantics><\/math><\/sup>SO(6) with the Binomial[15,5]=3003.<\/p>\n\n\n\n

G2 has 3 and E6 has 8 representations of orbit dimension 3003 (each shown below).<\/p>\n\n\n\n

The first column shows the G2 or E6 Coxeter-Dynkin diagram. The second column is the Weyl orbit. The third column is the identified maximal subgroup elements in the 3003 rep. The fourth column shows the maximal subgroup content and the fifth column shows its Coxeter-Dynkin diagrams.<\/p>\n\n\n\n

Please forgive (ignore) my use of filled instead of ringed nodes. The relevant information is the subgroup content in the third column.<\/p>\n\n\n\n

\"\"\/<\/a>
G2 3003 SU3 Rep<\/figcaption><\/figure>\n\n\n\n
\"\"\/<\/a>
G2 3003 SU2xSU2 Rep<\/figcaption><\/figure>\n\n\n\n
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G2 3003 SU2 Rep<\/figcaption><\/figure>\n\n\n\n
\"\"\/<\/a>
E6 3003 SO10xU1 Rep<\/figcaption><\/figure>\n\n\n\n
\"\"\/<\/a>
E6 3003 SU5xSU2 Rep<\/figcaption><\/figure>\n\n\n\n
\"\"\/<\/a>
E6 3003 SU33<\/sup> Rep<\/figcaption><\/figure>\n\n\n\n
\"\"\/<\/a>
E6 3003 G2 Rep<\/figcaption><\/figure>\n\n\n\n
\"\"\/<\/a>
E6 3003 SU3 Rep<\/figcaption><\/figure>\n\n\n\n
\"\"\/<\/a>
E6 3003 F4 Rep<\/figcaption><\/figure>\n\n\n\n
\"\"\/<\/a>
E6 3003 Sp8 Rep<\/figcaption><\/figure>\n\n\n\n
\"\"\/<\/a>
E6 3003 SU3xG2 Rep<\/figcaption><\/figure>\n\n\n\n
\"\"\/<\/a>
Annotated Binomial Collisions of the Pascal Triangle highlighting dimension 3003 Binomial[14, 6], Binomial[78, 2], and also Binomial[15, 4] which could be \u039b5<\/mn>5<\/annotation><\/semantics><\/math><\/sup> SO(6) = \u039b5<\/mn>5<\/annotation><\/semantics><\/math><\/sup> SU4<\/figcaption><\/figure>\n","protected":false},"excerpt":{"rendered":"

Please refer to the Baez blog post here. My first reaction is to question whether the “isomorphism of the irreducible representations” is or is not over-constraining the binomial’s numerical pattern match. I found a possible alternate path via \u039b55 SU(4) = \u039b55SO(6) with the Binomial[15,5]=3003. G2 has 3 and E6 has 8 representations of orbit … Continue reading Group Theoretic Binomial Collision Analysis<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-8351","post","type-post","status-publish","format-standard","hentry","category-physics"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t