{"id":8351,"date":"2026-09-21T18:44:21","date_gmt":"2026-09-22T01:44:21","guid":{"rendered":"https:\/\/theoryofeverything.org\/theToE\/?p=8351"},"modified":"2026-09-30T12:22:25","modified_gmt":"2026-09-30T19:22:25","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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List of collisions<\/figcaption><\/figure>\n<\/div>\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 (of the given orbit dimension)” is or is not over-constraining the binomial’s numerical pattern match.<\/p>\n\n\n\n

G2 has 3 and E6 has 8 representations of orbit dimension 3003 (each shown below). 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. Yet, these groups don’t have natural orbits of 3003 as E6 and G2 do. The two adjacent binomials in this 3003 triplet of collisions, namely Binomial[14,6] and Binomial[15,5], are part of the known infinite series of solutions with k=2 for binomial(n,m-1) = binomial(n-1,m) given by n = F_{2k}*F_{2k+1}; m = F_{2k-1}*F_{2k} where F_i is the i-th Fibonacci number.<\/p>\n\n\n
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Adjacent binomial collisions in the Infinite series o Fibonacci numbers<\/figcaption><\/figure>\n<\/div>\n\n\n

Below are the related maximal subgroups for G2 and E6. 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

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G2 3003 SU3 Rep<\/figcaption><\/figure>\n\n\n\n
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G2 3003 SU2xSU2 Rep<\/figcaption><\/figure>\n\n\n\n
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G2 3003 SU2 Rep<\/figcaption><\/figure>\n\n\n\n
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E6 3003 SO10xU1 Rep<\/figcaption><\/figure>\n\n\n\n
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E6 3003 SU5xSU2 Rep<\/figcaption><\/figure>\n\n\n\n
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E6 3003 SU33<\/sup> Rep<\/figcaption><\/figure>\n\n\n\n
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E6 3003 G2 Rep<\/figcaption><\/figure>\n\n\n\n
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E6 3003 SU3 Rep<\/figcaption><\/figure>\n\n\n\n
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E6 3003 F4 Rep<\/figcaption><\/figure>\n\n\n\n
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E6 3003 Sp8 Rep<\/figcaption><\/figure>\n\n\n\n
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E6 3003 SU3xG2 Rep<\/figcaption><\/figure>\n\n\n\n

The image below was created from Pane #2 Math: Number Theory: Mod 2-9 Pascal and Sierpinski Triangle<\/a>. It is used here to annotate the above binomial collisions which are all contained in the standard Pascal Triangle. <\/p>\n\n\n\n

This #2 pane (Pascal) was created for visualizing the Pascal and Sierpinski triangles, along with the Fibonacci and Lucas numbers.  In addition to allowing the change in size of the triangle, it highlights the binomial functions and allows the changing the modulus of the numbers used. The number backgrounds are colorized in the selected gradient. This pane was created with ideas from  Peter S. Williams\u2019 Mod 9 Pascal Triangle Physics http:\/\/naturalnumber.com\/<\/a><\/p>\n\n\n\n

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Annotated Binomial Collisions of the Pascal Triangle highlighting (blue Mod8=3) dimension 3003 in G2 Binomial[14, 6], E6 Binomial[78, 2] with k=2 Binomial[15, 5] which could be \u039b5<\/mn>5<\/annotation><\/semantics><\/math><\/sup> SO(6) = \u039b5<\/mn>5<\/annotation><\/semantics><\/math><\/sup> SU4 (albeit not a natural orbit of those groups). The other highlights are collisions of 120 (red Mod8=8), 210 (orange Mod8=2), 1540 (yellow Mod8=4), and 7140 (magenta Mod8=4). Two partial collisions (with one factor beyond this table <128) are in white rings as Binomial[19,5]=Binomial[153,2]=11628 (Mod8=4) and Binomial[17,8]=Binomial[221,2]=24310 (Mod8=6). I also added in the OEIS A003015<\/a> listed collision of Binomial[104,39]=Binomial[103,40]=61218182743304701891431482520 as black rings Mod8=8. As is the OEIS listed k=1, this k=3 collision should more properly be assigned as part of the infinite series of adjacent Fibonacci collisions such that it is not “unique” as the other 7 that are not exclusively related to the adjacent Fibonacci series.<\/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 (of the given orbit dimension)” is or is not over-constraining the binomial’s numerical pattern match. G2 has 3 and E6 has 8 representations of orbit dimension 3003 (each shown below). I found a possible … 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":[14],"class_list":["post-8351","post","type-post","status-publish","format-standard","hentry","category-physics","tag-math"],"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