Wythoff polytopes and low-dimensional homology of Mathieu groups

Sikirić, Mathieu Dutour; Ellis, Graham

doi:10.1016/j.jalgebra.2009.09.031

Cited by 10 publications

(5 citation statements)

References 16 publications

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“…where we have used the relation χ a + χ a ′ = 0 for all pairs of splitting representations. The characters for the pairs (7,8) and (16,17) are irrational for these conjugacy classes. For such pairs, rationality (and hence integrality) of the Fourier-Jacobi coefficients of the EOT Jacobi forms implies that ( N a (n) − N a ′ (n)) = 0 for (a, a ′ ) = (7, 8), (16,17).…”

Section: Implications Of Conjecture 21mentioning

confidence: 99%

“…The non-uniqueness of our solution for M 12 moonshine suggests we look for further constraints beyond the ones that we have imposed. It is known that like M 24 , M 12 has a non-trivial 3-cocycle [17]. One has H 3 (M 12 , Z) = Z 8 ⊕ Z 6 .…”

Section: There Is No Moonshine For M 12mentioning

confidence: 99%

“…The characters for the pairs (7,8) and (16,17) are irrational for these conjugacy classes. For such pairs, rationality (and hence integrality) of the Fourier-Jacobi coefficients of the EOT Jacobi forms implies that ( N a (n) − N a ′ (n)) = 0 for (a, a ′ ) = (7,8), (16,17). For all other pairs, equation (2.5) implies the weaker condition, ( N a (n) − N a ′ (n)) = 0 mod 2.…”

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confidence: 99%

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Two moonshines for L2(11) but none for M12

Govindarajan

Samanta

2019

Nuclear Physics B

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In this paper, we revisit an earlier conjecture by one of us that related conjugacy classes of M 12 to Jacobi forms of weight one and index zero. We construct Jacobi forms for all conjugacy classes of M 12 that are consistent with constraints from group theory as well as modularity. However, we obtain 1427 solutions that satisfy these constraints (to the order that we checked) and are unable to provide a unique Jacobi form. Nevertheless, as a consequence, we are able to provide a group theoretic proof of the evenness of the coefficients of all EOT Jacobi forms associated with conjugacy classes of M 12 : 2 ⊂ M 24 . We show that there exists no solution where the Jacobi forms (for order 4/8 elements of M 12 ) transform with phases under the appropriate level. In the absence of a moonshine for M 12 , we show that there exist moonshines for two distinct L 2 (11) sub-groups of the M 12 . We construct Siegel modular forms for all L 2 (11) conjugacy classes and show that each of them arises as the denominator formula for a distinct Borcherds-Kac-Moody Lie superalgebra.3. We address the non-existence of a moonshine for M 12 by providing two moonshines for L 2 (11) that arise as two distinct sub-groups of M 12 . Using this, we construct Siegel modular forms for all conjugacy classes using a product formula that arises naturally as a consequence of L 2 (11) moonshine. The modularity of this product is proven in two ways: (i) as an additive lift determined by the eta product, ηρ(τ ) -this works for all but three conjugacy classes (1 1 11 1 , 3 4 and 6 2 ), and (ii) as a product of rescaled Borcherds products -this works for all conjugacy classes.4. For all conjugacy classes of L 2 (11), we show the existence of Borcherds-Kac-Moody (BKM) Lie superalgebras for conjugacy classes of both the L 2 (11) by showing that the Siegel modular forms, ∆ρ k (Z), arise as their Weyl-Kac-Borcherds deominator identity. Figure 1 pictorially summarises the moonshines for L 2 (11).The plan of the manuscript is as follows. Following the introductory section, in section 2, we describe the conjecture 2.1 for M 12 moonshine. We find multiple solutions that satisfy all the constraints that are imposed. In proposition 2.4, we show that a stronger form of M 12 moonshine does not hold. We then show that there are unique solutions for two distinct L 2 (11) subgroups of M 12 . In section 3, we construct genus-two Siegel modular forms for all conjugacy classes for both

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Section: Implications Of Conjecture 21mentioning

confidence: 99%

Section: There Is No Moonshine For M 12mentioning

confidence: 99%

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confidence: 99%

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Two moonshines for L2(11) but none for M12

Govindarajan

Samanta

2019

Nuclear Physics B

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“…The low-degree homology groups of all Mathieu groups can be computed in HAP, and are listed in [14], where details of HAP's implementation are discussed. That paper was the first to compute H 3 (M 24 ), and was able to compute up to H 4 exactly for all Mathieu groups, and H 5 exactly for all Mathieu groups except M 24 , for which the 2-part was left ambiguous.…”

Section: Mathieu Groupsmentioning

confidence: 99%

“…With F 2 -coefficients, the entire cohomology rings of many of the smaller sporadic groups are listed in [2], and at large primes the cohomology rings of many sporadic groups are computed in [37,38]. The Mathieu entries are reviewed in [14]. Significantly, H 3 (M 24 ) was first computed in that paper using Graham Ellis's software package "HAP", which we have found can also determine H 3 (G) for G ∈ {HS, 2HS, J 2 , 2J 2 , J 1 , J 3 , McL} using the permutation models given in the ATLAS.…”

Section: Introductionmentioning

confidence: 99%

Third Homology of some Sporadic Finite Groups

Johnson-Freyd¹,

Treumann

2019

SIGMA

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Classification of Vertex-Transitive Zonotopes

Winter

2021

Discrete Comput Geom

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We give a full classification of vertex-transitive zonotopes. We prove that a vertex-transitive zonotope is a $$\Gamma $$ Γ -permutahedron for some finite reflection group $$\Gamma \subset {{\,\mathrm{O}\,}}(\mathbb {R}^d)$$ Γ ⊂ O ( R d ) . The same holds true for zonotopes in which all vertices are on a common sphere, and all edges are of the same length. The classification of these then follows from the classification of finite reflection groups. We prove that root systems can be characterized as those centrally symmetric sets of vectors, for which all intersections with half-spaces, that contain exactly half the vectors, are congruent. We provide a further sufficient condition for a centrally symmetric set to be a root system.

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Wythoff polytopes and low-dimensional homology of Mathieu groups

Cited by 10 publications

References 16 publications

Two moonshines for L2(11) but none for M12

Two moonshines for L2(11) but none for M12

Third Homology of some Sporadic Finite Groups

Classification of Vertex-Transitive Zonotopes

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