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How do I represent monomial reflection groups?

asked 2022-02-14 22:56:08 +0200

Zander K gravatar image

updated 2022-04-22 08:15:12 +0200

FrédéricC gravatar image

Let m and n be positive integers, let $C_m$ be the cyclic group of order m and B = $C_m$ × . . . × $C_m$ be the direct product of n copies of $C_m$. For each divisor p of m define the group A(m, p, n) by

A(m, p, n) = {($θ_1$, $θ_2$, . . . , $θ_n$) ∈ B | ($θ_1$$θ_2$ . . . $θ_n$)$^{m/p}$ = 1}.

It follows that A(m, p, n) is a subgroup of index p in B and the sym- metric group Sym(n) acts naturally on A(m, p, n) by permuting the coordinates. G(m, p, n) is defined to be the semidirect product of A(m, p, n) by Sym(n). It follows that G(m, p, n) is a normal subgroup of index p in the wreath product Cm ≀ Sym(n) and thus has order $m^nn!/p$. It is well known that these groups can be realized as finite subgroups of $GL_n(C)$, specifically as n × n matrices with exactly one non-zero entry, which is a complex m-th root of unity, in each row and column such that the product of the entries is a complex (m/p)th root of unity. Thus the groups G(m, p, n) are sometimes referred to as monomial reflection groups.

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Welcome to Ask Sage! Thank you for your question.

slelievre gravatar imageslelievre ( 2022-02-15 00:22:24 +0200 )edit
slelievre gravatar imageslelievre ( 2022-07-25 15:22:50 +0200 )edit

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answered 2022-02-15 05:42:40 +0200

rafael gravatar image

For a representation of a element w in a reflection group W you can type: w.to_matrix()

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Ok ok so I have gotten the tuples that generate the matrices, now how do I turn that into every possible matrix in the group?

Zander K gravatar imageZander K ( 2022-02-16 01:45:02 +0200 )edit

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Asked: 2022-02-14 22:56:08 +0200

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Last updated: Feb 14 '22