Deflation in Coxeter Groups G Eric Moorhouse based on recent work - - PowerPoint PPT Presentation

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Nov 03, 2023 494 likes •756 views

Deflation in Coxeter Groups G Eric Moorhouse based on recent work (1993-present) of John H. Conway and Christopher S. Simons The cube defines an infinite Coxeter group. The cube has four 6-cycles (as induced subgraphs). ~ Deflate every A 5

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Deflation in Coxeter Groups

G Eric Moorhouse

based on recent work (1993-present) of John H. Conway and Christopher S. Simons

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The cube defines an infinite Coxeter group.

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The cube has four 6-cycles (as induced subgraphs).

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Deflate every A5 to an A5 .

~

This can be done by imposing an additional relation s1s2s3s4s5s6s5s4s3s2 = 1 for every 6-cycle.

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Deflate every A5 to an A5 .

~

The resulting homomorphic image is a finite group isomorphic to PSp4(3)×2 = U4(2)×2

f order 51840.

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Let G be a finite group generated by a set of n elements of order 2. Then G is a homomorphic image of a Coxeter group defined by a graph on n vertices. But this presentation is not usually very concise. We would prefer n to be very small relative to |G|. Problem: Find groups G having a concise presentation obtained by deflating a Coxeter group. Examples grow on trees!

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The graph Y111

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Close A3 to form A3

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Goal: Close every 3-path to a 4-cycle, adding as few vertices as possible.

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Interesting examples arise also from:

Y222 (7 vertices) Yields the Petersen graph (10 vertices) Close induced 5-paths to 6-cycles Deflate every A6 to A6

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The resulting group is PSp4(3):2 = U4(2):2

f order 51840

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Interesting examples arise also from:

Y333 (10 vertices) Yields the incidence graph of PG2(2) (14 vertices) Close induced 7-paths to 8-cycles Deflate every A8 to A8

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The resulting group is the group O8

–(2):2

f order 394,813,440

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Interesting examples arise also from:

Y555 (16 vertices) Yields the incidence graph of PG2(3) (26 vertices) Close induced 11-paths to 12-cycles Deflate every A12 to A12

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The resulting group is the Bimonster (M×M):2

f order ≈ 1.31×10108

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Theorem (Ivanov-Norton, 1992) The Bimonster (M×M):2 is presented by the Coxeter group of the graph Y555 with the single additional relation (ab1c1ab2c2ab3c3)10 = 1.

a b1 b2 b3 c1 c2 c3

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V = R3k+1 is the set of all vectors with the Lorentzian inner product

V =

v11 v12 … v1k v21 v12 … v2k v31 v32 … v3k v∞ v•w = v11w11 + v12w12 + … + v3kw3k – v∞w∞ Vertices are represented by vectors v1,v2,…,vn such that vi•vj = 2. Vertices i,j are joined whenever vi•vj = ±1 (but possibly for other nonzero values of vi•vj as well).

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0 … 0 1 0 … 0 1 0 … 0 1 1

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… 1 –1 … … 0 … 0 1 0 … 0 1 0 … 0 1 … … 1 –1 … … … … 1 –1 1

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… 1 –1 … … 0 … 0 1 0 … 0 1 0 … 0 1 … … 1 –1 … … … … 1 –1 … … 1 –1 … … 1 –1 … … … … … 1 –1 1

The graphs Ykkk

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MAPLE demonstrations Y222 Y333 Y555

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