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table of marks

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Definition

Definition

(Burnside marks and Burnside character)

Let GG be a finite group and let XGSet finX \in G Set^{fin} any finite G-set. For [H][H] the conjugacy class of a subgroup HGH \subset G, the [H][H]-mark on XX is the cardinality of the set of HH-fixed points, hence the natural number

ϕ [H](X)|X H| \phi_{[H]}(X) \;\coloneqq\; \left\vert X^{H} \right\vert

of elements xXx \in X such that for each hHGh \in H \subset G we have h(x)=xh(x) = x.

This construction extends to a ring homomorphism

(1)ϕ : A(G) ϕ Conj(G) [X] ([H]|X H|) \array{ \phi &\colon& A(G) &\overset{\phi}{\longrightarrow}& \mathbb{Z}^{Conj(G)} \\ && [X] &\mapsto& \big( [H] \mapsto \left\vert X^H\right\vert\big) }

from the Burnside ring to the ring of tuples of integers of length the number Conj(G)Conj(G) of conjugacy classes of subgroups of GG.

This morphism is also called the Burnside character or mark homomorphism.

Remark

(marks in terms of homs)

Equivalent the set of [H][H]-marks of a G-set XX (Def. ) is the hom-set in GSet from G/HG/H to XX.

Proof

This follows from a basic standard argument. For completeness, we make it explicit:

By transitivity of the action on G/HG/H a GG-equivariant function f:G/HXf \colon G/H \to X is fully specified by its image f([e])Xf([e]) \in X of the equivalence class [e]G/H[e] \in G/H of the neutral element. Since this [e]G/H[e] \in G/H is fixed precisely by the elements in HGH \subset G it may, again by GG-equivariance, be mapped to any HH-fixed point f([e])X HXf([e]) \in X^H \subset X.

Of particular interest are the marks of the transitive G-sets, i.e. those isomorphic to sets G/HG/H of coset, for HGH\subset G a subgroup. These arrange into a table of marks:

Definition

(table of marks)

The table of Burnside marks (or table of marks, for short) of a finite group GG is the matrix indexed by conjugacy classes [H][H] of subgroups HGH \subset G whose ([H i],[H j])([H_i], [H_j])-entry is the [H j][H_j]-marks of G/H iG/H_i (Def. ), hence the number of fixed points of the action of H jH_j on the coset space G/H iG/H_i:

M ij|(G/H i) H j|. M_{i j} \;\coloneqq\; \left\vert \left(G/H_i\right)^{H_j} \right\vert \,.

(e.g. Pfeiffer 97, chapter The Burnside Ring and the Table of Marks)

Remark

(table of marks in terms of homs)

The expression of marks in terms of homs (Remark ) means here that the table of marks (Def. ) is equivalently given by

M ij|(G/H i) H j|=|Hom GSet(G/H j,G/H i)|. M_{i j} \;\coloneqq\; \left\vert \left(G/H_i\right)^{H_j} \right\vert \;=\; \left\vert Hom_{G Set}(G/H_j, G/H_i)\right\vert \,.


Properties

Relation to characters

The following Prop. that the Burnside character plays the same role for finite G-sets as characters of representations play for finite-dimensional linear representations, in that it faithfully reflects GG-sets. In fact the marks of a GG-set over cyclic subgroups coincides with the character of its permutation representation over any ground field (Prop. ) below.

Proposition

(Burnside character is injective)

The Burnside character (1) is injective. Hence any two finite G-sets are isomorphic precisely if they have the same Burnside marks (Def. ).

(e.g. tomDieck 79, Prop. 1.2.2, tomDieck 09, Prop. 5.1.1)

Proposition

(mark homomorphism on cyclic groups agrees with characters of corresponding permutation representations)

For SGSet finS \in G Set_{fin} a finite G-set, for kk any field and k[S]Rep k(G)k[S] \in Rep_k(G) the corresponding permutation representation, the character χ k[S]\chi_{k[S]} of the permutation representation at any gGg \in G equals the Burnside marks (Def. ) of SS under the cyclic group gG\langle g\rangle \subset G generated by gg:

χ k[S](g)=|X g|k. \chi_{k[S]}\big( g \big) \;=\; \left\vert X^{\langle g \rangle} \right\vert \;\in\; \mathbb{Z} \longrightarrow k \,.

Hence the mark homomorphism (Def. ) of GG-sets restricted to cyclic subgroups coincides with the characters of their permutation representations.

This statement immediately generalizes from plain representations to virtual representations, hence to the Burnside ring.

(e.g. tom Dieck 09, (2.15))

Proof

By definition of character of a linear representation, we have that

χ k[S](g)=tr k[S](g) \chi_{k[S]}(g) = tr_{k[S]}(g)

is the trace of the linear endomorphism k[S]gk[S]k[S] \overset{g}{\to} k[S] of the given permutation representation.

Now the canonical kk-linear basis for k[S]k[S] is of course the set SS itself, and so

χ k[S](g) =sS{1 | g(s)=s 0 | otherwise =|S g| =|S g| \begin{aligned} \chi_{k[S]}(g) & = \underset{ s \in S }{\sum} \left\{ \array{ 1 &\vert& g(s) = s \\ 0 &\vert& \text{otherwise} } \right. \\ & = \left\vert S^g \right\vert \\ & = \left\vert S^{\langle g \rangle} \right\vert \end{aligned}

Here in the first step we spelled out the definition of trace in the canonical basis, and in the second step we observed that the fixed point set of a cyclic group equals that of any one of its generating elements.

Relation to Burnside product

We discuss, in Prop. below, how the table of marks encodes the product in the Burnside ring of the given finite group GG. For this purpose we first consider two Lemma: Lemma and Lemma .

Lemma

(linear order on set of conjugacy classes of subgroups)

There exists a linear order lin\leq_{lin} on the set of conjugacy classes [H][H] of subgroups of GG such that a subgroup inclusion HHH \subset H' implies that [H] lin[H][H] \leq_{lin} [H'].

Proof

This follows by the general existence of linear extensions of partial orders applied to the subgroup lattice of GG.

Lemma

(table of marks is lower triangular invertible matrix)

With respect to any linear order on the conjugacy classes of subgroups as in Lemma , the table of marks (Def. ) becomes a lower triangular matrix over the integers with non-zero entries on the diagonal. In particular, it is an invertible matrix.

Proof

That the subgroup H jGH_j \subset G has any fixed points in G/H iG/H_i means that there is a gGg \in G such that hgH i=gH ih g H_i = g H_i for all hH jh \in H_j, and thus that g 1H jgg^{-1}H_{j}g is a subgroup of H iH_{i}, or in other words, that H jH_{j} is conjugate to a subgroup of H iH_{i}. Hence

(M ij>0)([H j] lin[H i]), \big( M_{i j} \gt 0 \big) \;\Rightarrow\; \big( [H_j] \leq_{lin} [H_i] \big) \,,

and thus the matrix MM is lower triangular.

Since at least H=1 G/HH = 1_{G/H} is fixed by HH, we moreover have that the diagonal entries are non-zero.

In the following, given a G-set G/H iG/H_i we write [G/H i]A(G)[G/H_i] \in A(G) for its isomorphism class, regarded as an element in the Burnside ring.

Definition

(Burnside multiplicities)

Given a choice of linear order on the conjugacy classes of subgroups of GG (for instance as in Lemma ), we say that the corresponding structure constants of the Burnside ring (or Burnside multiplicities) are the natural numbers

n ij n_{i j}^\ell \;\in\; \mathbb{N}

uniquely defined by the equation

(2)[G/H i]×[G/H j]=n ij [G/H ]. [G/H_i] \times [G/H_j] \;=\; \underset{ \ell }{\sum} n_{i j}^\ell [G/H_\ell] \,.
Proposition

(Burnside ring product in terms of table of marks)

The Burnside ring structure constants (n ij )\left( n_{i j}^\ell\right) (Def. ) are equal to the following algebraic expression in the table of marks (M ij)\left( M_{i j}\right) and its inverse matrix ((M 1) rs)\left( \left(M^{-1}\right)_{r s} \right) (which exists by Lemma ):

n ij =mM imM jm(M 1) m n_{i j}^\ell \;=\; \underset{m}{\sum} M_{i m} \cdot M_{j m} \cdot (M^{-1})_{m \ell}
Proof

Let tt be the dimension of MM, i.e. MM is a t×tt \times t matrix. For any 1mt1 \leq m \leq t, we compute as follows:

1tn ij M m =1tn ij |Hom GSet(G/H m,G/H )| =|Hom GSet(G/H m,n ij G/H )| =|Hom GSet(G/H m,G/H i×G/H j)| =|Hom GSet(G/H m,G/H i)||Hom GSet(G/H m,G/H j)| =M imM jm \begin{aligned} \underset{1 \leq \ell \leq t}{\sum} n_{i j}^\ell \cdot M_{\ell m} & = \underset{1 \leq \ell \leq t}{\sum} n_{i j}^\ell \cdot \left\vert Hom_{GSet} \big( G/H_m , G/H_\ell \big) \right\vert \\ & = \left\vert Hom_{GSet} \big( G/H_m , \underset{\ell}{\sum} n_{i j}^\ell \cdot G/H_\ell \big) \right\vert \\ & = \left\vert Hom_{GSet}\big( G/H_m, \; G/H_i \times G/H_j \big) \right\vert \\ & = \left\vert Hom_{GSet}\big( G/H_m, \; G/H_i \big) \right\vert \cdot \left\vert Hom_{GSet}\big( G/H_m, \; G/H_j \big) \right\vert \\ & = M_{i m} \cdot M_{j m} \end{aligned}

Here the third step uses the defining equation (2) of the structure constants n ij n_{i j}^\ell, while all other steps use that the mark homomorphism is a ring homomorphism, which we made manifest by expressing the marks via hom-sets (Remark ).

Thus we have that (n ij 1,n ij 2,,n ij t)\left( n_{i j}^{1}, n_{i j}^{2}, \ldots, n_{i j}^{t} \right) is a solution to the following system of equations.

M i1M j1 =M 11x 1++M t1x t M i2M j2 =M 12x 1++M t2x t M itM jt =M 1tx 1++M ttx t \begin{aligned} M_{i 1} \cdot M_{j 1} &= M_{1 1}x_{1} + \cdots + M_{t 1}x_{t} \\ M_{i 2} \cdot M_{j 2} &= M_{1 2}x_{1} + \cdots + M_{t 2} x_{t} \\ &\vdots \\ M_{i t} \cdot M_{j t} &= M_{1 t}x_{1} + \cdots + M_{t t} x_{t} \end{aligned}

But, since MM is invertible, the unique solution to this system of equations is given by the product of M 1M^{-1} and the transposition of (M i1M j1,,M itM jt)\left( M_{i 1} \cdot M_{j 1}, \ldots, M_{i t} \cdot M_{j t} \right). The claim follows immediately.

Corollary

The table of marks of a finite group determines its Burnside ring. That is to say, if the tables of marks of a pair of groups G 1G_{1} and G 2G_{2} are equal, then the Burnside ring of G 1G_{1} is isomorphic to the Burnside ring of G 2G_{2}.

Proof

The Burnside ring of a finite group is a free abelian group on the set G/H 1,,G/H tG / H_1, \ldots, G / H_t, where H 1,,H tH_1, \ldots, H_t are representatives of the conjugacy classes of that group, equipped with a certain multiplication. Thus it suffices to check that the structure constants of the Burnside rings coincide, which is established by the previous proposition.

Applications

References

The concept was introduced in

Textbook accounts and lecture notes include

Computer implementation is discussed in

See also

  • Götz Pfeiffer, The Subgroups of M 24M_{24}, or How to Compute the Table of Marks of a Finite Group, Experiment. Math. 6 (1997), no. 3, 247–270 (doi:10.1080/10586458.1997.10504613, web)

  • Liam Naughton, Götz Pfeiffer, Computing the table of marks of a cyclic extension, Math. Comp. 81 (2012), no. 280, 2419–2438.

  • Brendan Masterson, Götz Pfeiffer, On the Table of Marks of a Direct Product of Finite Groups, Journal of Algebra Volume 499, 1 April 2018, Pages 610-644 (arXiv:1704.03433)

Last revised on February 20, 2019 at 07:49:22. See the history of this page for a list of all contributions to it.