Lift of fractional D-brane charge to equivariant Cohomotopy theory

TL;DR

This paper investigates the lift of fractional D-brane charges to equivariant Cohomotopy theory, showing it constrains charges to rational values and resolves the irrational charge paradox in string theory.

Contribution

It explicitly computes the constraints on fractional D-brane charges imposed by lifting from equivariant K-theory to Cohomotopy, confirming the hypothesis that this lift excludes irrational charges.

Findings

The Boardman homomorphism maps the Burnside ring to the representation ring via permutation representations.

For certain finite groups, the comparison map surjects onto the sub-lattice of non-irrational characters.

The results support Hypothesis H by showing the lift excludes irrational D-brane charges.

Abstract

The lift of K-theoretic D-brane charge to M-theory was recently hypothesized to land in Cohomotopy cohomology theory. To further check this "Hypothesis H", here we explicitly compute the constraints on fractional D-brane charges at ADE-orientifold singularities imposed by the existence of lifts from equivariant K-theory to equivariant Cohomotopy theory, through Boardman's comparison homomorphism. We check the relevant cases and find that this condition singles out precisely those fractional D-brane charges which do not take irrational values, in any twisted sector. Given that the possibility of irrational D-brane charge has been perceived as a paradox in string theory, we conclude that Hypothesis H serves to resolve this paradox. Concretely, we first explain that the Boardman homomorphism, in the present case, is the map from the Burnside ring to the representation ring of the singularity group given by forming virtual permutation representations. Then we describe an explicit algorithm that computes the image of this comparison map for any finite group. We run this algorithm for binary Platonic groups, hence for finite subgroups of SU(2); and we find explicitly that for the three exceptional subgroups and for the first few cyclic and binary dihedral subgroups the comparison morphism surjects precisely onto the sub-lattice of the real representation ring spanned by the non-irrational characters.

Equations

$$\beta$$

$$\left(G{\mathrm{Set}}^{\mathrm{fin}}{/}_{\sim },\bigsqcup ,\times \right)=\left(G{\mathrm{Rep}}_{{\mathbb{F}}_1}^{\mathrm{fin}}{/}_{\sim },{\oplus }_{{\mathbb{F}}_1},{\otimes }_{{\mathbb{F}}_1}\right)\text{and}\left(G{\mathrm{Rep}}_k^{\mathrm{fin}}{/}_{\sim },{\oplus }_k,{\otimes }_k\right),$$

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$$1\simeq k[\ast ].$$

$$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 35.63132pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-35.63132pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\left(G\mathrm{Set}^{\mathrm{fin}}/_{\sim},\sqcup,\times\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 54.58792pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{k[-]}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 89.63132pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 59.63132pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 89.63132pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\left(G\mathrm{Rep}^{\mathrm{fin}}/_{\sim},\oplus,\otimes\right)}$}}}}}}}\ignorespaces}}}}\ignorespaces.$$

$$(V,V)\sim 0\phantom{AAA}\mbox{for all $V\in G\mathrm{Rep}^{\mathrm{fin}}_{k}/_{\sim}$}\,.$$

$$R_k(G):=K\left(G{\mathrm{Rep}}_k^{\mathrm{fin}}{/}_{\sim },\oplus ,\otimes \right).$$

$$(S,S)\sim 0\phantom{AA}\mbox{for all $S\in G\mathrm{Set}^{\mathrm{fin}}$}.$$

$$A(G):=K\left(G{\mathrm{Set}}^{\mathrm{fin}}{/}_{\sim },\bigsqcup ,\times \right).$$

$$A(G)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$R_{k}^{\mathrm{int}}(G)\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces R_{k}(G)$$

$${\chi }_{{}_V}:g⟼{\chi }_{{}_V}(g)\in \mathbb{Z}\subset k.$$

$$\beta\;:\;\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 14.57014pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-14.57014pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{A(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 19.20157pt\raise 6.79286pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.40398pt\hbox{$\scriptstyle{\beta^{\mathrm{int}}_{k}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 38.57014pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 38.57014pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{R^{\mathrm{int}}_{k}(G)\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 77.3472pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 101.34718pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 101.34718pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{R_{k}(G)}$}}}}}}}\ignorespaces}}}}\ignorespaces.$$

$$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 17.99966pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-17.99966pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{R^{\mathrm{int}}_{k}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 24.27744pt\raise 4.28406pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.28406pt\hbox{$\scriptstyle{=}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 41.99966pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 41.99966pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{R^{\mathrm{rat}}_{k}(G)\subset R_{k}(G)}$}}}}}}}\ignorespaces}}}}\ignorespaces\phantom{AAA}\mbox{for $\mathbb{Q}\subset k$}\,.$$

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$$A(G)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

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$$|G|=p^n,$$

$$[g]:=gH.$$

$$1\simeq k[\ast ]\in G{\mathrm{Rep}}_k^{\mathrm{fin}}$$

$$V^G\simeq \mathrm{Hom}(1,V)\in {\mathrm{Vect}}_k.$$

$$[G/H_1]=[G/H_2]⟺[H_1]=[H_2].$$

$$\underset{\genfrac{}{}{0ex}{}{[H]}{H\subset G}}{\oplus }\mathbb{Z}[H]\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$k[G/H_i]\otimes k[G/H_j]\simeq \underset{\ell }{\oplus }{n}_{ij}^{\ell }k[G/H_{\ell }].$$

$$M_{ij}:=\sum _{\ell }{n}_{ij}^{\ell }.$$

$$m_{ij}\;:=\;\left|\big{(}G/H_{i}\big{)}^{H_{j}}\right|\;\in\;\mathbb{Z}\,.$$

$$\big{(}H_{i}\subset H_{j}\big{)}\;\Rightarrow\;\big{(}[H_{i}]\leq[H_{j}]\big{)}\,.$$

$$\Big{(}M_{ij}=\big{|}\left(G/H_{i}\right)^{G_{j}}\big{|}\;>\;0\Big{)}\;\Rightarrow\;\big{(}[H_{j}]\leq[H_{i}]\big{)}\,,$$

$$n_{ij}^{\ell }=\sum _k{m}_{ik}\cdot m_{jk}\cdot (m^{-1}{)}^{k\ell }.$$

$$m_{ij}\;=\;\left|\ \mathrm{Hom}_{G\mathrm{Set}}\big{(}G/H_{j},G/H_{i}\big{)}\right|\,.$$

Full text

Concretely, we first explain that the Boardman homomorphism, in the present case, is the map from the Burnside ring to the representation ring of the singularity group given by forming virtual permutation representations. Then we describe an explicit algorithm that computes the image of this comparison map for any finite group. We run this algorithm for binary Platonic groups, hence for finite subgroups of ${\rm SU}(2)$; and we find explicitly that for the three exceptional subgroups and for the first few cyclic and binary dihedral subgroups the comparison morphism surjects precisely onto the sub-lattice of the real representation ring spanned by the non-irrational characters.

1. 1 Fractional brane charge quantization in M-theory

2. 2 The equivariant Boardman homomorphism $\beta$

3. 2.1 The Burnside ring and K-theory

4. 2.2 Analysis of the image of $\beta$

5. 2.3 An algorithm for the image of $\beta$

6. 3 The image of $\beta$ – Examples

7. 3.1 Cyclic groups: $C_{n}$

8. 3.2 Binary dihedral groups: $2D_{2n}\simeq\mathrm{Dic}_{n}$

9. 3.3 Binary exceptional groups: $2T$, $2I$, $2O$

10. A Background on platonic groups and categorical algebra

11. A.1 The Platonic groups

12. A.2 Categorical algebra

We present here a curious computation in elementary representation theory (Theorem 3.1 below), the background for which we introduce in detail in section 2 below. Besides its mathematical content, which is of interest in itself as explained in section 2.2 below, we argue that this result impacts on open questions in the foundations of string theory, as we explain next in section 1.

1 Fractional brane charge quantization in M-theory

The issue of irrational D-brane charge. It is a long-standing conjecture [Wi98, Sec. 5.1] that the charge lattice of fractional D-branes [DGM97] stuck at $G$-orientifold singularities is the $G$-equivariant K-theory of the singular point, hence the representation ring of $G$ (e.g. [Gr05]). However, it was argued already in [BDH*+*02, 4.5.2] that not all elements of the representation ring can correspond to viable D-brane charges, and a rationale was sought for identifying a sub-lattice of physical charges. Independently, in [BDS00, (2.8)] it was highlighted that the possibility of irrational D-brane RR-charge is a “paradox” [BDS00] that needs to be resolved.

However, for fractional D-branes stuck at $G$-orbifold singularities, the RR-charge is rationally proportional to the character (recalled as Def. 2.28 below) of the corresponding representation (by [DGM97, (3.8)][BCR00, (4.65)][RS13, (4.102)]):

Orbifold

D-brane theory

D-brane charge

at $G$-singularity

mass

charge in

$g$-twisted sector

V\in\begin{array}[]{c}\mathrm{KO}_{G}\\ \begin{rotate}{90.0}\small!\simeq\end{rotate}\\ R_{{}_{\mathbb{R}}}(G)\end{array} \begin{array}[]{c}\mathrm{dim}(V)\\ \begin{rotate}{90.0}\small!=\end{rotate}\\ \chi_{{}_{V}}(e)\end{array} \begin{array}[]{c}\mathrm{tr}_{{}_{V}}(g)\\ \begin{rotate}{90.0}\small!=\end{rotate}\\ \chi_{{}_{V}}(g)\end{array}

Representation

theory

linear

$G$-representation

character value

at neutral element

character value

at element $g\in G$

Table 1 – The translation between fractional D-brane charge at $G$-orbifold singularities and characters of linear $G$-representations.

In view of [BDS00] this means that representations with irrational characters would reflect physically spurious fractional D-brane charges, even though they do appear in equivariant K-theory. Several authors tried to find a resolution of the paradox of possible irrational D-brane charge [Ta00][Zh01][Ra02] but the situation has remained inconclusive.

The open problem of formulating M-theory. Of course, perturbative string theory, where this paradox is encountered, is famously supposed to be just a limiting case of an elusive non-perturbative theory with working title M-theory (see e.g.[Du99][HSS18, Sec. 2]). It is to be expected that the full M-theory implies constraints not seen from the string perturbation series. This is directly analogous to the now popular statement that perturbative string theory, in turn, implies constraints not seen in effective quantum field theory, separating the “Landscape” of effective field theories that do lift to perturbative string vacua from the “Swampland” of those that do not [Va05], indicated on the right of Figure 1. However, despite the tight web of hints on its limiting cases, actually formulating M-theory remains an open problem (see [NH98, p. 2][Mo14, Sec. 12][CP18, p. 2][HSS18, Sec. 2]).

Hypothesis H – M-brane charge quantization in Cohomotopy. But recent analysis of the structure of the bouquet of super $p$-brane WZ terms from the point of view of homotopy theory ([Sa18][FSS16a][FSS15][BSS18], see [FSS19a] for review) has suggested a new hypothesis about the precise nature of the brane charge structure in M-theory. This Hypothesis H [FSS19b][FSS19c] asserts that the generalized cohomology theory which quantizes brane charge in M-theory, in analogy to how twisted K-theory is expected to quantize D-brane charge in string theory (see [BSS18, Sec. 1][GS19] for pointers), is unstable Cohomotopy cohomology theory [Bo36][Sp49][Pe56], specifically $J$-twisted Cohomotopy [FSS19b, Def. 3.1]. On flat orbifold spacetimes this is unstable equivariant Cohomotopy [SS19, (2)][HSS18], traditionally considered in its stabilized approximation [Seg71][Ca84][Lu05]. This is indicated on the left of Figure 1, which contextualizes our proposal and provides a global perspective:

[TABLE]

**Figure 1: Relations between M-brane charges in M-theory via Cohomotopy and D-brane charges in string theory via K-theory. **

The Boardman homomorphism to equivariant K-theory. Displayed on the left of Figure 1* is the Boardman homomorphism [Ad74, II.6], which, under Hypothesis H, is the M-theoretic analog of the Chern character (that maps D-brane charge in string theory to flux forms in effective field theory) now mapping brane charge in non-perturbative M-theory to its perturbative approximation in string theory. As discussed in detail in [SS19, 3.1.2], classical results in equivariant homotopy theory (see [Bl17]) identify [Seg71][tD79, 8.5.1][Lu05, 1.13] the stable equivariant Cohomotopy of a $G$-orbifold singularity with the Burnside ring – a non-linear combinatorial analog of the $G$-linear representation ring. Furthermore, under this identification, the Boardman homomorphism to equivariant K-theory identifies with the homomorphism that linearizes a combinatorial $G$-set (of M-branes) to a linear $G$-representation (of D-brane Chan-Paton factors), see Figure 2. We lay out the relevant definitions in full detail in section 2 below. As explained in [SS19], we have the following picture:*

[TABLE]

\pi^{V}_{{}_{G}}\big{(}\big{(}\mathbb{R}^{V}\big{)}^{\mathrm{cpt}}\big{)} $\Sigma^{\infty}$

stabilization

[TABLE]

$\mathbb{S}_{G}^{0}$ $\beta$

Boardman

homomorphism

$\!\!\simeq$

$A_{G}$

[TABLE]

\underset{\mbox{ \tiny\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1} linearization }}{\footnotesize\mathbb{R}[-]}

[TABLE]

$\mathrm{KO}_{G}^{0}$

$\!\!\!\!\simeq$

$\mathrm{RO}(G)$

[TABLE]

\overset{}{\mbox{ \tiny e.g. one O^{{}^{-}}!!-plane and two branes }}$$\overbrace{\phantom{AAAAAAAAAAAAAAAAAAA}} \overset{}{\mbox{ \tiny\begin{tabular}[]{c}minus the trivial G$-set\\ with two regular$G-sets\end{tabular} }}$$\overbrace{\phantom{AAAAAA}} \overset{}{\mbox{ \tiny\begin{tabular}[]{c}minus the trivial G$-representation\\ plus two times the regular$G-representation\end{tabular} }}$$\overbrace{\phantom{AAAAAA}} \infty$$\infty$$\infty$$\infty$$+\mathbf{1}_{{}_{\mathrm{triv}}} $-\mathbf{4}_{{}_{\mathrm{reg}}}$ $-\mathbf{4}_{{}_{\mathrm{reg}}}$

Figure 2 – Virtual $G$-representations of fractional branes classified by equivariant Cohomotopy* in the vicinity of ADE-singularities, seen through the Boardman homomorphism to equivariant K-theory, according to [SS19]. Shown is a situation for $G=\mathbb{Z}_{4}$.*

With this explicit form of the equivariant Boardman homomorphism at $G$-singularities ($\beta\simeq\mathbb{R}[-]$) in hand, we obtain here a concrete computational handle on the implications of Hypothesis H, which may serve as further tests of this hypothesis.

Fractional D-brane charge at orientifold singularities under Hypothesis H.* In [SS19] we had laid out in detail how, under Hypothesis H, equivariant Cohomotopy measures charges of M5-branes stuck at ADE-orientifold singularities, and how this translates, under the Boardman homomorphism, to charges in equivariant K-theory of fractional D-branes stuck at these singularities. The main result of [SS19] is that the lift of fractional D-brane charge on orientifolds from equivariant K-theory through the Boardman homomorphism to equivariant Cohomotopy (as in Figure 2) implies expected anomaly cancellation conditions for D-branes in orientifolds, the RR-field tadpole cancellation conditions.*

Approach: Numerical fractional D-brane charge under Hypothesis H.* Here we investigate further implications of this lift of brane charge from K-theory to Cohomotopy: Using the explicit form of the Boardman homomorphism explained below in section 2.1, we develop in section 2.3 an effective algorithm for explicitly computing the image of the Boardman homomorphism as a sublattice in the representation ring, and hence in the equivariant K-theory ring of fractional D-brane charges. This allows us to concretely read off which fractional D-brane charges lift to M-theory under Hypothesis H, as indicated in Figure 1.*

Physics result.* Our main result, Theorem 3.1 below, establishes that, in all the examples of singularity groups which we compute, the image of the Boardman homomorphism in the real representation ring (appropriate for orientifold charges) consists precisely of the integral characters; these are equivalently the non-irrational characters (by Prop. 2.3 below). Under the translation of Table 1 this means that Hypothesis H removes precisely the irrational D-brane charges from the charge lattice of fractional D-branes stuck at ADE-singularities. Hence, in as far as irrational D-brane charge is a paradox [BDS00][Ta00][Zh01][Ra02]**, Hypothesis H resolves this paradox.*

We explain now in section 2 how these issues are formulated mathematically, translating the question of lifting D-brane charge to Cohomotopy to a question purely in representation theory.

2 The equivariant Boardman homomorphism $\beta$

2.1 The Burnside ring and K-theory

Finite group actions control orbifold spacetime singularities in string theory. For $G$ a finite group one may consider linear as well as purely combinatorial actions of $G$ on some set (see e.g. **[Dr71]**[tD79]**[Ke99]**[LP12]**). We will be concerned with the relation between these two types of actions (see e.g. **[tD79]**[Be91]**[Bo10]**).

Basics.* Traditionally, for $k$ any field, the linear actions receive more attention as they are the $k$-linear representations of $G$, namely the group homomorphisms $G\to\mathrm{Aut}_{k}(V)$ from $G$ to the $k$-linear invertible maps from a given $k$-vector space $V$ to itself. More elementary than this concept is that of plain $G$-sets, which are instead group homomorphism of the form $G\to\mathrm{Aut}(S)$, from $G$ to all the invertible functions from some set $S$ to itself (permutations).*

To emphasize that these two concepts, while clearly different, are conceptually related, one may appeal to the lore of the “field with one element” $\mathbb{F}_{1}$ **[Ti56]**[KS]**[So04]**[CCM09]**[Man08]** (see **[Th16]**[Lo18]** for recent surveys) and regard plain sets as vector spaces over $\mathbb{F}_{1}$, and plain set-theoretic permutations as being the $\mathbb{F}_{1}$-linear maps $\mathrm{Aut}(S)=\mathrm{Aut}_{\mathbb{F}_{1}}(S)$.

In any case, for every finite group $G$ and field $k$, the isomorphism classes of finite nonlinear actions and of finite-dimensional linear actions of $G$ form two rings-without-negatives,

[TABLE]

where addition is given by disjoint union of $G$-sets and by direct sum of $G$-representations, respectively, while the product operation is given by Cartesian product of $G$-sets and by tensor product of $G$-representations, respectively.

We will be interested in a canonical comparison map from $G$-sets to $G$-representation: By forming $k$-linear combinations of elements of a finite set, every $G$-set G\;\;\;\raisebox{-1.0pt}{\begin{rotate}{90.0} \curvearrowright\end{rotate}}\;S in $A(G)$ spans a $k$-linear representation

[TABLE]

These are called the permutation representations. The archetypical example is the regular representation $k[G]$, which is the linearization of $G$ acting on its own underlying set by group multiplication from the left. A simple but important example is the trivial 1-dimensional representation $\mathbf{1}$ of $G$, which is the linearization of the point, regarded as a $G$-set:

[TABLE]

The construction (2) of permutation representations from $G$-sets is clearly linear and multiplicative, in that it extends to a homomorphism between the above rings-without-negatives (1)

[TABLE]

While this establishes the canonical comparison map between nonlinear and linear $G$-actions, there is nothing much of interest to be said about it.

Passage to K-theory.* This situation changes drastically as soon as we consider not just plain $G$-sets and $G$-representations, but also their “anti-$G$-sets” and “anti-$G$-representations”, namely as we group-complete the rings-without-negatives in (1) to actual rings, by adjoining additive inverses for all elements. Concretely, a virtual $G$-representation is represented by a pair $(V^{+},V^{-})$ of two $G$-representations, thought of as a plain $G$-representation $V^{+}$ and an anti-$G$-representation $V^{-}$; and the K-group completion is obtained by quotienting out from the evident group (ring) that these virtual representation/anti-representation pairs form the equivalence relation*

[TABLE]

This can viewed as exhibiting pair-creation/annihilation of bound states of a representation with its own anti-representation. The resulting ring is called the representation ring of $G$, denoted

[TABLE]

An analogous construction applies to virtual $G$-sets represented by pairs consisting of a $G$-set and an anti-$G$-set, subject to the relation of pair-creation/annihilation of bound states of a $G$-set $S$ with its anti-$G$-set:

[TABLE]

Now the resulting ring is known as the Burnside ring of $G$ (see e.g. **[So67]**[Dr71]**[tD79]**[Ke99]**), denoted

[TABLE]

While plain permutation representations (2) generally form just a very small subset of all isomorphism classes of $k$-linear representations, the passage to virtual permutation representations drastically changes the picture: Since every plain permutation representation decays into a direct sum of irreducible linear representations, the formal difference of two permutation representations in a virtual permutation representation may partially cancel out to become equal, in the representation ring, to a representation that is not itself a plain permutation representation.

For example, in the simplest non-trivial case, where $G=C_{2}$ is the finite group of order 2, the 1-dimensional alternating representation $\mathbf{1}_{\mathrm{alt}}$ is clearly not a permutation representation itself. But it is a direct summand in the regular representation $k[C_{2}]=\mathbf{1}+\mathbf{1}_{\mathrm{alt}}$. Since the other summand is a permutation representation, $\mathbf{1}=k[\ast]$, the alternating representation may then be isolated as the formal difference $\mathbf{1}_{\mathrm{alt}}\;=\;k[C_{2}]-k[\ast]$, thus as a virtual permutation representation.

The comparison morphism.* Therefore, while the step from plain to virtual actions and representations is small, it has drastic consequences, as it potentially reduces much of linear representation theory to pure combinatorics. In order to quantify this effect, one observes that the construction (4) of permutation representations evidently extends linearly to virtual $G$-sets and virtual $G$-representations, to a homomorphism*

[TABLE]

from the Burnside ring (6) to the representation ring (5), taking virtual $G$-sets to virtual $G$-representations.

We may associate to the representation theory of $G$ over $k$ the following interpretation.

[TABLE]

Intuition might suggest that generally the cokernel of $\beta$ is large, while the kernel of $\beta$ is generally small. This is indeed the case for the restriction (4) of $\beta$ to actual (as opposed to virtual) $G$-sets and $G$-representations. However, the inclusion of anti-$G$-sets and anti-representations and passage to the K-groups of virtual $G$-sets and virtual $G$-representations completely changes the picture. It turns out that the kernel of $\beta$ almost never vanishes: in characteristic zero its rank is the difference of the number of non-cyclic by cyclic subgroups of $G$ (Prop. 2.7 below). At the same time, classical results give that the cokernel of $\beta$ often vanishes: for instance, over $k=\mathbb{Q}$ it vanishes for all cyclic groups (Prop. 2.6 below), as well as for all $p$-groups (**[Seg72]**, recalled as Prop. 2.8 below), while for $G=S_{n}$ a symmetric group, the cokernel of $\beta$ vanishes even for all fields $k$ of characteristic zero (Prop. 2.9 below).

This is a remarkable state of affairs, which deserves further investigation.

The mathematical incarnation of the goals and results stated at the end of section 1 are the following:

Goal.* Here our goal is to give explicit descriptions of the cokernel of $\beta$, over the rational, real and complex numbers, for further concrete examples of finite groups $G$. We are particularly interested in the case of the binary Platonic groups, namely the finite subgroups of $\mathrm{SU}(2)$ (recalled in section A.1).*

Method.* In section 2.3 we describe an algorithm for the image of $\beta$, Theorem 2.37 below. In section 3 we apply this algorithm and compute the cokernel of $\beta$ in various concrete examples, Theorem 3.1. In appendix A we collect some background material.*

Results.* The table in Theorem 3.1 shows that in all examples computed here, notably for the three exceptional finite subgroups of $\mathrm{SU}(2)$ as well as the seven first cases of binary dihedral groups, the image of $\beta$ in the real representation ring consists precisely of the sub-lattice of integer characters, hence (by Prop. 2.3) of non-irrational characters. The same holds true for larger classes of Examples which we have computed, but are not showing here. We conjecture that it holds true for all binary dihedral groups.*

2.2 Analysis of the image of $\beta$

In order to set the scene for the considerations below, we record some well-known general facts about the image of the comparison morphism $\beta$ (7).

Definition 2.1** (Sub-lattice of integer-valued characters).**

For $G$ a finite group and $k$ a field, write

[TABLE]

for the sub-lattice of the representation ring given by those representations $V\in R_{k}(G)$ whose characters $\chi_{{}_{V}}$ (Def. 2.28) are integer-valued

[TABLE]

Proposition 2.2** ($\beta$ takes values in integer-valued characters).**

The comparison morphism $\beta$ (7) has its image inside the integer-valued characters (Def. 2.1), hence it factors as

[TABLE]

Proof.

By Example 2.29. ∎

The following is elementary, but important:

Proposition 2.3** (e.g. [Na-cycl]).**

If the ground field $k$ has characteristic zero, then a character (Def. 2.28) that takes values in the rational numbers $\mathbb{Q}\subset k$ in fact already takes values in the integers $\mathbb{Z}\subset\mathbb{Q}\subset k$. Hence if $R_{k}^{\mathrm{rat}}(G)\subset R_{k}(G)$ denotes the sublattice of rational-valued characters, in analogy to the sub-lattice of integer-valued characters in Def. 2.1, then these sub-lattices are in fact equal:

[TABLE]

In particular, if the ground field $k=\mathbb{Q}$ is itself the rational numbers, then all characters are integer-valued characters (Def. 2.1), hence in this case the canonical inclusion (8) is an isomorphism:

[TABLE]

Proof.

In general, characters are cyclotomic integers. Over the rationals the only cyclotomic integers are the actual integers. ∎

Remark 2.4** (Factorizations of the comparison map $\beta$).**

In summary, Prop. 2.2 and Prop. 2.3 say that we have the following commuting diagram of factorizations of the morphism $\beta$ (7) that sends $G$-sets to their linear permutation representation:

[TABLE]

Hence it is worthwhile to first record what is known about the image of $\beta$ over $\mathbb{Q}$;

Proposition 2.5** (e.g. [tDi09, proof of Prop. 4.5.4]).**

Over the rational numbers, $k=\mathbb{Q}$, the image of $A(G)\overset{\beta}{\to}R_{\mathbb{Q}}(G)$ (7) is at least a sub-lattice of full rank (i.e. has the same number of generators as $R_{\mathbb{Q}}(G)$). This full-rank sublattice is spanned by the permutation representations (2) of the form $\mathbb{Q}[G/C_{i}]$ for $C_{i}\subset C_{n}$ ranging over the cyclic subgroups.

Proposition 2.6** (e.g. [tDi09, Example (4.4.4)]).**

For $G=C_{n}=\mathbb{Z}/n$ a cyclic group and $k=\mathbb{Q}$ the rational numbers, $\beta$ is an isomorphism

[TABLE]

Proposition 2.7**.**

The only finite groups $G$ for which $A(G)\xrightarrow{\beta_{k}}R_{k}(G)$ is injective over $k=\mathbb{Q}$ (hence over $k=\mathbb{R},\mathbb{C}$) are the cyclic groups $G=C_{n}$.

Proof.

We know that a linear basis for $A(G)$ is given by the cosets $G/H$ for $H$ ranging over conjugacy classes of all subgroups of $G$, while a linear basis for $R_{\mathbb{Q}}(G)$ is given by the isomorphism classes of irreducible $\mathbb{Q}$-linear representations. But the latter are in bijection to just the cyclic subgroups of $G$ (e.g. [tDi09, Prop. 4.5.4] ). This means that when $G$ is not itself cyclic, then the cardinality of a linear basis for $A(G)$ is strictly larger than the cardinality of a linear basis for $R_{\mathbb{Q}}(G)$, so that no morphism $A(G)\to R_{\mathbb{Q}}(G)$ can be injective. On the other hand, when $G$ is a cyclic group then $\beta$ is an isomorphism by Prop. 2.6, and hence in particular injective. ∎

Less immediate is the following result:

Proposition 2.8** ([Seg72]).**

If the finite group $G$ is a $p$-group, hence if its number of elements is the $n$th power $p^{n}$ of some prime number $p$ by some natural number $n\in\mathbb{N}$

[TABLE]

then over $k=\mathbb{Q}$ the comparison morphism \textstyle{A(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{R_{\mathbb{Q}}(G)} (7) is surjective.

The standard representation theory of symmetric groups in terms of Young diagrams and Specht modules yields the following statement:

Proposition 2.9** (e.g. [Dr86, Section 3]).**

*Over any ground field $k$ of characteristic zero, and for $G=S_{n}$ any symmetric group of permutations of $n\in\mathbb{N}$ elements, the comparion map

\textstyle{A(S_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{k}(S_{n})} (7) is surjective.*

For some other classes of finite groups, formulas for the cokernel and kernel of $\beta$ are known; see, e.g., **[BD16]**.

2.3 An algorithm for the image of $\beta$

We describe here an algorithm for computing the image and cokernel of $\beta$ (7). The end result is Theorem 2.37 below. Establishing the algorithm involves only elementary representation theory (see **[Dr71]**[tD79]**[Be91]**[Ke99]** **[Rob06]**[Bo10]**[LP12]**) and basic monoidal category theory (see **[Mc65]**[Bo94]**) but seems to be new.

Throughout, $G$ is a finite group and $k$ is a field.

Definition 2.10** (Category of $G$-sets).**

We write $G\mathrm{Set}^{\mathrm{fin}}$ for the category of finite sets equipped with $G$-action, called $G$-sets, for short.

This is a symmetric monoidal category (**[Bo94, vol 2, 6.1]**) with respect to Cartesian product of $G$-sets, which is given by the plain Cartesian product of underlying sets, equipped with the diagonal $G$-action. For example, the underlying set of the group $G$ becomes a $G$-set by the left multiplication action of $G$ on itself. More generally, for $H\subset G$ any subgroup, the set $G/H$ of cosets is still a $G$-set by the left action of $G$ on itself. The elements of $G/H$ are equivalence classes of elements $g$ of $G$, often denoted $gH$, for which we will write

[TABLE]

Generally, we use square brackets to indicate the equivalence classes or isomorphism classes. In particular we write $[G/H]$ for the isomorphism class of the $G$-set $G/H$ as an object of $G\mathrm{Set}$.

Definition 2.11** (Category of $k$-linear $G$-representations).**

We write $G\mathrm{Rep}^{\mathrm{fin}}_{k}$ for the category of finite dimensional $k$-linear $G$-representations.

This is a symmetric monoidal category (**[Bo94, vol 2, 6.1]**) with respect to the standard tensor product of representations, which we denote simply by “$\otimes$”.

Definition 2.12** (The trivial irrep).**

We write

[TABLE]

for the trivial 1-dimensional $G$-representation (3), equivalently the permutation representation (2) of the singleton $G$-set. This is the tensor unit for the tensor monoidal structure on $G\mathrm{Rep}_{k}$: For $V\in G\mathrm{Rep}_{k}$ any representation, the hom-space out of $\mathbf{1}$ into $V$ is the vector space $V^{G}$ of $G$-invariants in $V$, hence of elements which are fixed by $G$:

[TABLE]

Definition 2.13** (The irreducible $G$-sets).**

The action of $G$ on a $G$-set $S$ is called transitive if for all pairs of elements $s_{1},s_{2}\in S$ there exists a group element that takes them into each other: $gs_{2}=g_{2}$.

Every transitive $G$-set $S$ is isomorphic to a set of cosets $G/H$, equipped with the canonical $G$-action induced from the left action of $G$ on itsef, where $H$ is isomorphic to the stabilizer subgroup $\mathrm{Stab}(s)\subset G$ of $s$ in $S$. Two such $G$-sets of cosets are isomorphic, $G/H_{1}\simeq G/H_{2}$, precisely if $H_{1}$ and $H_{2}$ are conjugate to each other, as subgroups of $G$. If we denote isomorphism classes of $G$-sets by square brackets, and also denote conjugacy classes of subgroups by square brackets, then this means that

[TABLE]

Consequently, we have the following.

Proposition 2.14** (Canonical linear basis for Burnside ring).**

Every finite $G$-set is a disjoint union of such transitive $G$-sets (Def. 2.13). Hence the abelian group underlying the Burnside ring is the free abelian group on elements $[G/H]$, one for each conjugacy class $[H]$ of subgroups of $H$:

[TABLE]

We would like to get a handle on the following object:

Definition 2.15** (Multiplicities multiplication table of the Burnside ring).**

Let \big{\{}[H_{i}]\big{\}}_{i} be an indexing of the set of conjugacy classes $[H]$ of subgroups $H\subset G$.

1. (i)

The structure constants of the Burnside ring $A(G)$ is the set of natural numbers $\{n_{ij}^{\ell}\}$ defined by

[TABLE] 2. (ii)

The total multiplicities table of the Burnside ring $A(G)$ is the quadratic matrix whose $(i,j)$-entry is

[TABLE]

Before discussing the crucial role of the total multiplicities (11) for our purpose, (to which we come in Prop. 2.22 below) we first record an efficient way of computing them:

Definition 2.16** (e.g [Pf97, Def. 1.1]).**

Given a finite group $G$, its table of marks is the square matrix $m$ indexed by the conjugacy classes $[H]$ of subgroups $H\subset G$ whose $[H_{i}],[H_{j}]$-entry is the number of fixed points of the $H_{j}$-action on $G/H_{i}$

[TABLE]

Proposition 2.17** (Ordering).**

There exists a linear ordering $\leq$ of the set of conjugacy classes of subgroups of $G$ which extends the inclusion relation of subgroups, in that

[TABLE]

With respect to any such linear ordering, the table of marks $m$ (Def. 2.16) is a lower triangular matrix with positive entries on its diagonal, hence in particular an invertible matrix.

Proof.

First of all, inclusion of subgroups defines a partial order and every partial order extends to a linear order. (For our finite ordered set this follows immediately by induction, splitting off a minimal element in each step; more generally see [Mar30].) Then, observe that $H_{j}$ having any fixed points on $G/H_{i}$ means that it is conjugate to a subgroup of the stabilizer group of $[e]\in G/H_{i}$. But the latter is manifestly $H_{i}$ itself. Hence

[TABLE]

which says that $m$ is lower triangular. Finally, it is clear that at least $[e]\in G/H_{i}$ is fixed by $H_{i}$, hence that the diagonal entries are positive. ∎

Proposition 2.18** (Multiplicities explicitly).**

The Burnside multiplicities $m_{ij}^{\ell}$ (Def. 2.15) are given by the following algebraic expression in terms of the entries of the table of marks $m$ (Def. 2.16) and its inverse matrix $m^{-1}$ (from Prop. 2.17):

[TABLE]

Proof.

Notice that the entry $m_{ij}$ of the table of marks may equivalently be thought of as the cardinality of the set of homomorphism from the $G$-set $G/H_{j}$ to the $G$-set $G/H_{i}$:

[TABLE]

(Because, by transitivity of the action, any such homomorphism is determined by its image of $[e]\in G/H_{j}$ and by $G$-equivariance this has to be sent to any $H_{j}$-fixed point of $G/H_{i}$.) With this we compute as follows:

[TABLE]

Here in the third step we used the defining equation (10) of the Burnside multiplicities $n_{ij}^{\ell}$, and otherwise we used evident properties of sets of homomorphisms. Now matrix multiplication of both sides of this equation with the inverse matrix $m^{-1}$ yields the claimed relation. ∎

In order to understand the meaning of the total multiplicities (11), we consider now some basic facts, all elementary.

Proposition 2.19** (Self-duality of permutation representations).**

If $\mathrm{char}(k)\neq|H|$, then the permutation representation $k[G/H]$ (2) is a dualizable object in the symmetric monoidal representation category $(G\mathrm{Rep}_{k},\otimes)$ (e.g. [Bo94, vol 2, 6.1]) and is in fact self-dual:

[TABLE]

Proof.

We need to find morphisms

[TABLE]

in $G\mathrm{Rep}_{k}$ that make the following triangle commutes (the “triangle identity”, recalled in section A.2):

[TABLE]

where we are notationally suppressing the unitors and associators, as usual. With $[g]\in G/H\subset k[G/H]$ denoting both the equivalence class of an element $g\in G$ as well as the corresponding basis element of $k[G/H]$, we claim that the following choice works:

[TABLE]

Here the fraction on the left makes sense by assumption on the characteristic of $k$. Also, it is immediate that these linear maps do respect the $G$-action and hence are morphisms in $G\mathrm{Rep}_{k}$. Because with this, we check for every $[\tilde{g}]\in k[G/H]$ that:

[TABLE]

∎

As a direct consequence we obtain:

Proposition 2.20** (Internal hom between permutation representations).**

If $\mathrm{char}(k)\neq|H_{1}|$, then the internal hom between the permutation representation $k[G/H_{1}]$ and $k[G/H_{2}]$ (2) exists in $(G\mathrm{Rep}_{k},\otimes)$ and is given by the tensor product of representations:

[TABLE]

Proof.

Generally, if duals exist, the internal hom is given by

[TABLE]

With this, the statement follows by Prop. 2.19. ∎

Proposition 2.21** (Trivial irrep in transitive permutation representations).**

The permutation representation $k[G/H]$ of a transitive $G$-set (Def. 2.13) contains precisely one direct summand of the trivial 1-dimensional representation (Def. 2.12):

[TABLE]

Proof.

By definition, every element $v\in k[G/H]$ is a formal linear combination of cosets $[g]$:

[TABLE]

for coefficients $v_{[g]}\in k$. If there exists $[g_{1}],[g_{2}]\in G/H$ such that $v_{[g_{1}]}\neq v_{[g_{2}]}$ then, by transitivity of the $G$-action, there exist $g\in G$ with $g[g_{1}]=[g_{2}]$. But since the $[g]$ constitute a basis of $k[G/H]$, this implies that $gv\neq v$, hence that $v$ is not $G$-invariant. Therefore, the only $G$-invariant vectors $v\in k[G/H]$ are those all whose coefficients $v_{[g]}$ agree. These clearly form a 1-dimensional subspace. ∎

As a corollary we obtain:

Proposition 2.22** (Multiplication table via invariants and via hom-spaces).**

The entries $M_{ij}:={\sum}_{\ell}\,n_{ij}^{\ell}$ in the table of multiplication multiplicities in the Burnside ring (Def. 2.15) are equivalently

1. (i)

the dimensions of the subspaces of $G$-invariants in the Burnside products; 2. (ii)

the dimensions of the external homs of the two given basis elements;

[TABLE]

Proof.

The first equality follows from Prop. 2.21 applied to the Definition 2.15 of the structure constants, which gives the following isomorphism

[TABLE]

The second equality comes from the following sequence of isomorphisms

[TABLE]

Here the first equivalence expresses a general relation between external and internal homs, via the tensor unit (Def. 2.12), the second is from (2.12) and the last one is Prop. 2.20. ∎

Now it is useful to relate this to Schur’s Lemma. For this purpose it turns out to be convenient to think in terms of the following inner product.

Definition 2.23** (Inner product on Burnside ring).**

For $V_{1},V_{2}\in G\mathrm{Rep}_{k}$, write

[TABLE]

for the dimension of the vector space of representation homomorphism between them. By $\mathbb{Z}$-linearity this extends to a $\mathbb{Z}$-valued pairing on the Burnside ring:

[TABLE]

In terms of this pairing, Schur’s Lemma says the following:

Lemma 2.24** (Schur’s Lemma).**

The pairing $\langle-,-\rangle$ from Def. 2.23 is a symmetric and $\mathbb{Z}$-bilinear inner product on the abelian group underlying the representation ring $R_{k}(G)$. With respect to this inner product, the set of isomorphism classes $\rho_{i}$ of irreducible representations of $G$

1. (i)

is always an orthogonal basis, where each basis element has positive norm-square; 2. (ii)

is even an orthonormal basis if the field $k$ is algebraically closed.

Using this perspective, we amplify the following:

Remark 2.25** (Matching of multiplicities and independence).**

The statement of Prop. 2.22 is that, in terms of the inner product $\langle-,-\rangle$ from Def. 2.23, the multiplicities multiplication table coincides with the table of inner products of Burnside-basis elements:

[TABLE]

Also notice that Prop. 2.22 implies that the $k$-linear dimension of $k$-linear hom-spaces between $k$-linear permutation representations of transitive $G$-sets is independent of the ground field $k$:

[TABLE]

since the multiplication multiplicities matrix $M$ of the Burnside ring is manifestly independent of $k$.

We now use this to discuss explicit matrix representations of $\beta$.

Definition 2.26** (Upper triangular form of the Burnside multiplicities matrix).**

For $G$ a finite group, let

[TABLE]

be an integral upper triangular form (e.g. [GP90, p. 3,4]) of the Burnside multiplication multiplicities matrix (Def. 2.15), hence with $U\in\mathrm{GL}(N,\mathbb{Z})$ an invertible matrix whose left-multiplication implements row reduction on $M$ ($N$ the number of conjugacy classes of subgroups of $G$). Write

[TABLE]

for the result of deleting the zero-rows from (14). Write

[TABLE]

for the corresponding permutation representations and write

[TABLE]

for their norm-square with respect to the inner product (Def. 2.23).

Proposition 2.27** (Matrix for $\beta$).**

Consider the upper triangular form $\tilde{H}$ of the Burnside multiplication matrix, Def. 2.26.

(i)* Then the corresponding permutation representations $V_{i}\in R_{k}(G)$ (16) are orthogonal, in that their inner products (Def. 2.23) satisfy*

[TABLE]

and they linearly span the image of $\beta$: $\left\langle V_{i}\right\rangle_{i}\;\simeq\;\mathrm{im}(\beta)\subset R_{k}(G)$.

(ii)* Specifically, the matrix that represents $\beta$ with respect to the basis of the $G/H_{j}\in A(G)$ and the basis of $V_{i}\in\mathrm{Im}(\beta)\subset R_{k}(G)$ is*

[TABLE]

(In particular this means that the $i$th row $\tilde{H}_{i\bullet}$ of $\tilde{H}$ is divisible by $d_{i}$.)

(iii)* Hence for every subgroup $H_{j}\subset G$, the image of $\beta$ on the corresponding $G$-set $G/H_{j}$ is the following linear combination of the representations $V_{i}$ (16), from Def. 2.26:*

[TABLE]

Proof.

The first statement follows from [PT91]: By (13) we have that $M=A^{T}\cdot A$ is a positive semi-definite matrix of inner products, where $A$ is the matrix whose columns are the permutation representations $k[G/H]$ expanded in terms of the irreps of $G$. By [PT91, top of p. 5] this implies that the same row reduction which turns $M$ into upper-triangular form takes $A$ to a matrix whose non-vanishing columns $V_{i}$ constitute an orthogonal basis of the linear span of the $k[G/H]$.

The remaining statement just spells this out by immediate computation:

[TABLE]

Here the first line is by Def. 2.26, and in the second step we used Prop. 2.20 in the inner product notation from Def. 2.23 (as in Remark 2.25). In the third step we used the linearity of the inner product from Prop. 2.24, in the fourth step we inserted the definition of $V_{i}$ (16). In the fifth step we just identified $\beta(H_{j})=k[G/H_{j}]$, for emphasis. Finally we used the assumption (18) to identify the coefficient $\beta_{ij}$ of $V_{i}$ in $\beta(H_{j})$. ∎

With a linear basis for the image of $\beta$ thus in hand, it just remains to express it in a form that may directly be compared to standard classifications available from the linear representation theory of finite groups. For completeness, recall:

Definition 2.28** (Characters).**

For $G$ a finite group and $k$ a field, a function from the underlying set of $G$ to $k$ is called a class function if it is constant on conjugacy classes of $G$, hence if it factors as

[TABLE]

Hence class functions form a $k$-vector space of dimension the number of conjugacy classes in $G$:

[TABLE]

For $V\in R_{k}(G)$ a representation, the map sending any $g\in G$ to its trace, when regarded as a linear map $V\overset{g}{\to}V$ via this representation, is a class function (by basic properties of the trace), called the character $\chi_{V}$ of $V$:

[TABLE]

The following example is immediate but important:

Example 2.29** (Character of permutation representation).**

For $S\in A(G)$ a finite $G$-set, the character (Def. 2.28) of its permutation representation $k[S]$ (2) is the function that sends $g$ to the number of elements in $S$ that are fixed by the given action of $g$:

[TABLE]

The relevance of characters is that, in characteristic zero, they already completely characterize linear representations, while being more manifestly tractable:

Proposition 2.30** (e.g. [tDi09, Theorem 2.2.5]).**

If the field $k$ is of characteristic zero, then the map that sends a $k$-linear $G$-representation to its character (Def. 2.28) is an injection of the $k$-vector space of isomorphism classes of finite-dimensional $G$-representations into the vector space (20) of class functions (Def. 2.28)

[TABLE]

If $k$ is, in addition, a splitting field for $G$ (notably if $k=\mathbb{C}$ is the complex numbers), then this map is even an isomorphism.

As usual, it is convenient to organize this data in character tables. In order to make our list of examples in section 3 be unambiguously intelligible, we briefly dwell on the notation for character tables.

Definition 2.31** (Character table).**

For $(W_{i}\in R_{k}(G))_{i\in\{1,\cdots,\}}$ a tuple of (possibly virtual) $k$-linear representations of a finite group $G$, their character table is the $n\times{|\mathrm{ConjCl}(G)|}$-matrix with values in $k$ whose $(i,j)$-th entry is the value $\chi_{{}_{W_{i}}}(g_{j})$ of the character $\chi_{{}_{W_{i}}}$ (Def. 2.28) on any element $g_{j}$ of the $j$th conjugacy class $[g_{j}]\in\mathrm{ConjCl}(G)$.

Example 2.32** (Irreducible character table over $\mathbb{C}$).**

By Prop. 2.30 the characters of irreducible representations over $k=\mathbb{C}$ the complex numbers form a linear basis of the representation ring $R_{\mathbb{C}}(G)$. We will denote these irreducible representations by $(\rho_{i}\in R_{\mathbb{C}})(G)$ and will display the corresponding character table (Def. 2.31) as follows (conjucagy classes being labeled by the order of their elements):

[TABLE]

The character tables of irreducible representations over the complex numbers, for many finite groups of small order, have been tabulated in the literature, for instance in [Dok-GroupNames].

Example 2.33** (Irreducible character table over $\mathbb{R}$ ).**

For $k\subset\mathbb{C}$ a subfield, under the ring homomorphism of “extension of scalars”

[TABLE]

the values of characters to not change. Hence if $k$ is of characteristic zero, $\mathbb{Q}\subset k\subset\mathbb{C}$, then, by Prop. 2.30, we may equivalently express the character of any linear representation $W\in R_{k}(G)$ over $k$ after tensoring it with $\mathbb{C}$. This is a linear combination of the complex irreducible characters $\chi_{{}_{\rho_{i}}}$ from above

[TABLE]

Therefore, when $\mathbb{Q}\subset k\subset\mathbb{C}$ and for $(\cdots,W\in R_{k}(G),\cdots)$ a tuple of $k$-linear representations, we may and will express the corresponding character table as a table of linear combinations of the irreducible complex characters:

[TABLE]

In this fashion we will in particular state the irreducible character tables over $k=\mathbb{R}$ the real numbers, which may again be found in the literature for many finite groups of small order.

Specifically, the complex character tables available in the literature (e.g. **[Dok-GroupNames]**) list the type of the corresponding complex representation, from which the character table of irreducible representations over the real numbers may be extracted (or conversely, as in **[Mon-Reps]**), via the following basic fact:

Proposition 2.34** (e.g. [Rob06, p. 4]).**

Let $G$ be a finite group, and consider the complexification map on the representation ring $R_{\mathbb{R}}(G)\xrightarrow{(-)\otimes_{\mathbb{R}}\mathbb{C}}R_{\mathbb{C}}(G)$. Then every irreducible complex representation $V\in R_{\mathbb{C}}(G)$ is of exactly one of the following three types, depending on how it arises as a direct summand of an irreducible real representation $W\in R_{\mathbb{R}}(G)$:

[TABLE]

In this fashion we may now identify the image of $\beta$ via the character table of its basis elements:

Proposition 2.35** (Character table of linear basis of image of $\beta$).**

The character (Def. 2.28) of a basis element $V_{i}$ (16) of the image of $\beta$ (Prop. 19) is the class function given by

[TABLE]

where on the right we have the sum over conjugacy classes $H_{\ell}$ of subgroups of $G$ of the product of the entries of the base change matrix from (15) with the number of elements in $G/H_{\ell}$ that are fixed by the action of $g$.

Proof.

This follows by Example 2.29. ∎

Hence, in conclusion, we have the following.

Proposition 2.36** (Recognizing surjectivity of $\beta$).**

For $k$ of characteristic zero, let $(\rho_{i}\in R_{k}(G))$ be the irreducible $k$-linear representations, spanning the representation ring $R_{k}(G)$. Then the comparison morphism (7) is surjective precisely if the corresponding tuple of characters $\chi_{{}_{\rho_{i}}}$ (Def. 2.28) is related to the set of characters ${\chi}_{{}_{V_{i}}}$ in Def. 2.35 by an invertible integer matrix

[TABLE]

Proof.

Since, by construction, the $V_{j}$ (16) are linearly independent and span the image of $\beta$ (Prop. 2.27) and since the $\rho_{i}$ span $R_{k}(G)$, the number of the $V_{j}$ is smaller or equal to the number of $\rho_{i}$, hence the number must be equal if $\beta$ is surjective. This means that for surjectivity there must be an invertible integer matrix relating the $(V_{j})$ to the $(\rho_{i})$. But by Prop. 2.30 this is the case precisely if there is such a matrix relating the characters of these representations. ∎

This concludes our algorithmic description of the image of $\beta$. To summarize, the algorithm proceeds as follows. Here

• •

$N:={|\mathrm{ConjCl}(G)|}$* is the number of conjugacy classes of $G$;*

• •

$r:=\mathrm{rank}(\mathrm{image}(\beta))$* is the rank of the image of $\beta$ (the number of $V_{j}$);*

• •

$n$* is the number of isomorphism classes of irreducible representations $\rho_{i}$.*

Theorem 2.37** **(Algorithm for the cokernel of $\beta$).

Let $G$ be a finite group and $k$ a field of characteristic zero.

1. (1)

Extract from standard literature:

• •

the character table of irreducible $k$-linear representations (Examples 2.32, 2.33)

[TABLE] 2. (2)

Compute:

1. (a)

the multiplication table \big{(}n^{\ell}_{ij}\big{)} (10) of the Burnside ring, efficiently so via (12); 2. (b)

the resulting table of total multiplication multiplicities \left(M_{ij}\right):=\ \big{(}\underset{\ell}{\sum}n_{ij}^{\ell}\big{)} (11); 3. (c)

its upper triangular form $H:=U\cdot M$ (14); 4. (d)

the result $\tilde{H}=\tilde{U}\cdot M$ (15) of deleting its zero-rows; 5. (e)

the character table of the resulting linear basis for the image of $\beta$ (21)

[TABLE] 3. (3)

Read off the quotient of the lattice spanned by the vectors $\chi_{{}_{\rho_{i}}}$ by that spanned by the vectors $\chi_{{}_{V_{j}}}$:

[TABLE]

Remark 2.38** (Simplification in examples).**

(i) In all examples that we compute in section 3, upper triangular form of the multiplicities matrix in the third step of the algorithm of Theorem 2.37 is achieved by the most straightforward row reduction, where, incrementally in the row number, a suitable integer multiple of each row is subtracted from all those beneath it.

(ii) This is remarkable, since, in general, row reduction over the integers needs more and more intricate steps than this; see e.g. [GP90, p. 3–4]. That this happens is due to the fact that $M$ is a positive semidefinite matrix, as explained in [PT91].

(iii) Furthermore, in each case the resulting rows $V_{i}$ happen to be actual representations, as opposed to virtual representations. This makes our algorithm very efficient, and makes it easy to read off the image of $\beta$ in each example. It seems clear that this particularly nice behavior of row reduction on the Burnside multiplicities matrix is due to a very special property of the latter. It would be interesting to understand this phenomenon theoretically.

3 The image of $\beta$ – Examples

Given a finite group $G$ and its irreducible characters over a given field $k$ of characteristic zero, Theorem 2.37 provides an effective algorithm for identifying the image of $A(G)\xrightarrow{\;\beta\;}R_{k}(G)$ (7) and checking whether $\beta$ is surjective. We have implemented this algorithm in Python, available as an ancillary file. Here we spell out various example computations. In summary, we obtain the result shown in Theorem 3.1. In particular, the shaded entries show that over the real numbers $\beta$ has vanishing cokernel/is surjective onto the ring of integer characters (i.e. onto non-irrational characters, by Prop. 2.3).

Theorem 3.1** (Cokernel of $\beta$ for binary Platonic groups).**

The following table lists the cokernels

[TABLE]

of the permutation-representation morphism $A(G)\xrightarrow{\beta_{k}}R_{k}(G)$ (7) and its corestriction to the integer-valued character ring (Prop. 2.2, Remark 2.4), for finite subgroups of $SU(2)$ in the D- and E-series (via Prop. A.1) and some relatives, over ground fields $k\in\{\mathbb{Q},\mathbb{R},\mathbb{C}\}$:

[TABLE]

For emphasis we highlight by example how to read the table in Theorem 3.1:

• •

an entry “[math]” means that the cokernel vanishes, hence that $\beta$ is surjective;

• •

an entry “$\frac{\mathbb{Z}[\rho]}{\mathbb{Z}[2\rho]}$” means that the image of $\beta$ consists of all those virtual representations whose $\rho$-component has even multiplicity;

• •

an entry “$\frac{\mathbb{Z}[\rho_{1},\rho_{2}]}{\mathbb{Z}[\rho_{1}+\rho_{2}]}$” means that the image of $\beta$ consists of all those virtual representations whose $\rho_{1}$-component has the same weight as their $\rho_{2}$-component.

Here $\rho_{i}$ refers to the irreducible representations as tabulated in the respective subsection below. From the character tables given there one also reads off whether the character of $\rho_{i}$ is integer-valued or else (by Prop. 2.3) irrational. The cokernel for $\beta^{\mathrm{int}}_{k}$ is obtained from that of $\beta_{k}$ by removing those generators from the numerator that have irrational-valued characters.

3.1 Cyclic groups: $C_{n}$

For completeness, we include discussion of the first three cyclic groups, which may be thought of as completing the D-series of finite subgroups of $\mathrm{SU}(2)$ (from Prop. A.1) in the low range. These simple examples may serve to introduce and illustrate our notation for recording application of the algorithm (Theorem 2.37) in the examples to follow further below.

3.1.1 The cyclic group $C_{2}$

Group name: $C_{2}$ (**[Dok-$C_{2}$]**)

Group order: 2

Subgroups:

[TABLE]

Burnside ring product:

[TABLE]

Table of multiplicities:

[TABLE]

Upper triangular form:

[TABLE]

Character table for image of $\beta$:

[TABLE]

Character table of irreps:

over $\mathbb{C}$:

[TABLE]

over $\mathbb{R}$:

[TABLE]

Hence the cokernel of $\beta$ is:

[TABLE]

3.1.2 The cyclic group $C_{3}$

Group name: $C_{3}$ (**[Dok-$C_{3}$]**)

Group order: 3

Subgroups:

[TABLE]

Burnside ring product:

[TABLE]

Table of multiplicities:

[TABLE]

Upper triangular form:

[TABLE]

Character table for image of $\beta$:

[TABLE]

Character table of irreps:

over $\mathbb{C}$:

[TABLE]

over $\mathbb{R}$:

[TABLE]

Hence the cokernel of $\beta$ is:

[TABLE]

3.1.3 The cyclic group $C_{4}$

Group name: $C_{4}$ (**[Dok-$C_{4}$]**)

Group order: 4

Subgroups:

[TABLE]

Burnside ring product:

[TABLE]

Table of multiplicities:

[TABLE]

Upper triangular form:

[TABLE]

Character table for image of $\beta$:

[TABLE]

Character table of irreps:

over $\mathbb{C}$:

[TABLE]

over $\mathbb{R}$:

[TABLE]

Hence the cokernel of $\beta$ is:

[TABLE]

3.2 Binary dihedral groups: $2D_{2n}\simeq\mathrm{Dic}_{n}$

The binary dihedral groups have the following presentation (see e.g. **[Dok-$2D_{2n}$]** **[Li18]**):

[TABLE]

The order of $2D_{2n}$ is $4n$ and has $n+3$ conjugacy classes:

[TABLE]

The $n+3$ complex irreducible characters are given by:

[TABLE]

with $k=2,...,n-1$ and $\zeta=e^{2\pi i/2n}.$

The representations Triv. and 1A are always real. For $n$ even 1B and 1C are real. For $n$ odd 1A and 1B are complex, and 1A$+$1B is real irreducible. For $k$ even $\rho_{k}$ is real. For $k$ odd $\rho_{k}$ is quaternionic, and so $2\rho_{k}$ is real.

3.2.1 Binary dihedral group: $2D_{4}\simeq\mathrm{Dic}_{2}\simeq Q_{8}$

Group name: $2D_{4}\simeq Q_{8}$ (**[Dok-$2D_{4}$]**)

Group order: ${|2D_{4}|}=8$

Subgroups:

[TABLE]

Burnside ring product:

[TABLE]

Table of multiplicities:

[TABLE]

Upper triangular form:

[TABLE]

Character table for image of $\beta$:

[TABLE]

Character table of irreps **[Dok-$2D_{4}$, Mon-$Q_{8}$]**

over $\mathbb{C}$:

[TABLE]

over $\mathbb{R}$:

[TABLE]

Hence the cokernel of $\beta$ is:

[TABLE]

3.2.2 Binary dihedral group: $2D_{6}\simeq\mathrm{Dic}_{3}$

Group name: $2D_{6}$ (**[Dok-$2D_{6}$]**)

Group order: $|2D_{6}|=12$

Subgroups:

[TABLE]

Burnside ring product:

[TABLE]

Table of multiplicities:

[TABLE]

Upper triangular form:

[TABLE]

Character table for image of $\beta$:

[TABLE]

Character table of irreps **[Dok-$2D_{6}$]**

over $\mathbb{C}$:

[TABLE]

over $\mathbb{R}$:

[TABLE]

Hence the cokernel of $\beta$ is

[TABLE]

3.2.3 Binary dihedral group: $2D_{8}\simeq\mathrm{Dic}_{4}\simeq Q_{16}$

Group name: $2D_{8}$ (**[Dok-$2D_{8}$]**)

Group order: $|2D_{8}|=16$

Subgroups:

[TABLE]

Table of multiplicities:

[TABLE]

Upper triangular form:

[TABLE]

Character table for image of $\beta$:

[TABLE]

Character table of irreps **[Dok-$2D_{8}$]**:

over $\mathbb{C}$

[TABLE]

over $\mathbb{R}$

[TABLE]

Hence the cokernel of $\beta$ is:

[TABLE]

3.2.4 Binary dihedral group: $2D_{10}\simeq\mathrm{Dic}_{5}$

Group name: $2D_{10}$ (**[Dok-$2D_{10}$]**)

Group order: $|2D_{10}|=20$

Subgroups:

[TABLE]

Table of multiplicities:

[TABLE]

Upper triangular form:

[TABLE]

Character table for image of $\beta$:

[TABLE]

Character table of irreps **[Dok-$2D_{10}$]**

over $\mathbb{C}$

[TABLE]

over $\mathbb{R}$

[TABLE]

Hence the cokernel of $\beta$ is:

[TABLE]

3.2.5 Binary dihedral group: $2D_{12}\simeq\mathrm{Dic}_{6}$

Group name: $2D_{12}$ (**[Dok-$2D_{12}$]**)

Group order: $|2D_{12}|=24$

Subgroups:

[TABLE]

Table of multiplicities:

[TABLE]

Upper triangular form:

[TABLE]

Character table for image of $\beta$:

[TABLE]

Character table of irreps

over $\mathbb{C}$

[TABLE]

over $\mathbb{R}$

[TABLE]

Hence the cokernel of $\beta$ is

[TABLE]

3.2.6 Binary dihedral group: $2D_{14}\simeq\mathrm{Dic}_{7}$

Group name: $2D_{14}$

Group order: $|2D_{14}|=28$

Subgroups:

[TABLE]

Table of multiplicities:

[TABLE]

Upper triangular form:

[TABLE]

Character table for image of $\beta$:

[TABLE]

[TABLE]

3.2.7 Binary dihedral group: $2D_{16}\simeq\mathrm{Dic}_{8}\simeq Q_{32}$

Group name: $2D_{16}$

Group order: $|2D_{16}|=32$

Subgroups:

[TABLE]

Table of multiplicities:

[TABLE]

Upper triangular form:

[TABLE]

Character table for image of $\beta$:

[TABLE]

[TABLE]

3.3 Binary exceptional groups: $2T$, $2I$, $2O$

We discuss the three exceptional cases in the E-series of the finite subgroups of $\mathrm{SU}(2)$ (from Prop. A.1).

3.3.1 Binary tetrahedral group: $2T=\mathrm{SL}(2,3).$

Group name: $2T$ (**[Dok-$2T$]**)

Group order: $|2T|=24$

Subgroups:

[TABLE]

Table of multiplicities:

[TABLE]

Upper triangular form:

[TABLE]

Character table for image of $\beta$:

[TABLE]

Character table of irreps **[Dok-$2T$, Mon-$2T$]**:

over $\mathbb{C}$:

[TABLE]

over $\mathbb{R}$

[TABLE]

where $\zeta_{3}:=\tfrac{1}{2}\left(-1+\root \of{3}i\right)$

Hence the cokernel of $\beta$ is:

[TABLE]

3.3.2 Binary octahedral group: $2O\simeq\mathrm{CSU}(2,\mathbb{F}_{3}).$

Group name: $2O$ (**[Dok-$2O$]**)

Group order: ${|2O|}=48$

Subgroups:

[TABLE]

Table of multiplicities:

[TABLE]

Upper triangular form:

[TABLE]

Character table for image of $\beta$:

[TABLE]

Character table of irreps **[Dok-$2O$, Mon-$2O$]**

over $\mathbb{C}$

[TABLE]

over $\mathbb{R}$

[TABLE]

Hence the cokernel of $\beta$ is:

[TABLE]

3.3.3 Binary icosahedral group: $2I\simeq\mathrm{SL}(2,5).$

Group name: $2I$ (**[Dok-$2I$]**)

Group order: $|2I|=120$

Subgroups:

[TABLE]

Table of multiplicities:

[TABLE]

Upper triangular form:

[TABLE]

Character table for image of $\beta$:

[TABLE]

Character table of irreps **[Dok-$2I$]**

over $\mathbb{C}$

[TABLE]

over $\mathbb{R}$

[TABLE]

where $\phi:=\tfrac{1}{2}\left(1+\root \of{5}\right)$ is the golden ratio.

Hence the cokernel of $\beta$ is

[TABLE]

3.3.4 The general linear group: $\mathrm{GL}(2,\mathbb{F}_{3})$

Group name: $\mathrm{GL}(2,\mathbb{F}_{3})$ (**[Dok-$\mathrm{GL}(2,3)$*]*111 The representation theory of this group $\mathrm{GL}(2,\mathbb{F}_{3})$ is deceptively similar to that of the binary octahedral group $2O\simeq\mathrm{CSU}(2,\mathbb{F}_{3})$ (discussed Sect. 3.3.2): Both have the same character table over $\mathbb{C}$, the only difference being in the Schur indices, hence in the real character table. In fact, several online databases of character tables had misidentified the two groups, which became apparent when our computation of the image of $\beta$ revealed real representations of $2O$ that contradicted available character tables. We are indebted to James Montaldi for patiently double-checking computations with us and to Tim Dokchitser for swiftly recognizing and fixing the issue with the databases. )

Group order: $|\mathrm{GL}(2,\mathbb{F}_{3})|=48$

Subgroups:

[TABLE]

Table of multiplicities:

[TABLE]

Upper triangular form:

[TABLE]

Character table for image of $\beta$:

[TABLE]

Character table of irreps **[Dok-$\mathrm{GL}(2,3)$]**:

over $\mathbb{C}$

[TABLE]

over $\mathbb{R}$

[TABLE]

Hence the cokernel of $\beta$ is

[TABLE]

Appendix A Background on platonic groups and categorical algebra

To be reasonably self-contained, we briefly collect some background material which will be directly useful for our discussion and for connecting the physical and the mathematical sides of the arguments.

A.1 The Platonic groups

We are interested in the ADE groups as the main groups appearing in orbifolds in string theory. There are two variants, ones that are subgroups of the orthogonal group $\mathrm{SO}(3)$ and ones that are subgroups of the unitary group $\mathrm{SU}(2)$. Our focus will be on the latter.

Proposition A.1** (ADE-classification of the finite rotation groups [Kl84]).**

The finite subgroups of $\mathrm{SU}(2)$ are given, up to conjugacy, by the following classification (where $n\in\mathbb{N}$):

[TABLE]

Full proof for finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ is in **[MBD1916]**, recalled in **[Ser14, Sec. 2]**. Full proof for $\mathrm{SO}(3)$ is spelled out in **[Re05, Theorem 11]**; from this the case of $\mathrm{SU}(2)$ is given in **[Kee03, Theorem 4]**. See also **[Li18, Chapter 1,2]** for an elementary treatment.

Our discussion will require details on the structure of these finite groups and how they relate to each other. Hence we highlight the following pattern. The full subgroup lattice of ${\rm SU}(2)$ under the three exceptional finite subgroups from Prop. A.1 (using the subgroup lattices from **[Dok-$2T$, Dok-$2O$, Dok-$2I$]**).

[TABLE] \textstyle{\mathrm{SU}(2)}$$\textstyle{2O\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E\textstyle{2T\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2D_{8}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2D_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2D_{10}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}        $\textstyle{2D_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$D$\textstyle{2D_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$   $\textstyle{2D_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$    \textstyle{C_{10}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{8}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}   \textstyle{C_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}A\textstyle{C_{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}   $\textstyle{C_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$             \textstyle{C_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ast\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}

A.2 Categorical algebra

For ease of reference, we briefly recall the concept of internal homs in compact closed categories **[Bo94, vol 2, 6.1]**, also **[Mc65]**.

The categories $G\mathrm{Set}^{\mathrm{fin}}$ (Def. 2.10) and $G\mathrm{Rep}^{\mathrm{fin}}_{k}$ (Def. 2.11) enjoy various properties that are directly analogous to familiar properties of the category $\mathrm{Vect}^{\mathrm{fin}}_{k}$ of finite-dimensional $k$-vector spaces. The language of categorical algebra allows to make these analogies explicit, and such that one may reason uniformly in all three cases.

For $V,W\in\mathrm{Vect}^{\mathrm{fin}}_{k}$ two vector spaces, the set $\mathrm{Hom}(V,W)$ of linear maps (“homomorphisms”) $V\to W$ between them becomes itself canonically a vector space, by pointwise multiplication with $k$ and pointwise addition of values of functions. When we want to emphasize that we regard the set $\mathrm{Hom}(V,W)$ as equipped with this vector space structure, we write $[V,W]$ for it.

One way to make this vector space of linear functions $[V,W]$ more explicit is to consider the dual vector space $V^{\ast}$. With that in hand we have a canonical linear isomorphism

[TABLE]

which identifies the tensor product space of $V^{\ast}$ with $W$ as the vector space of linear maps from $V$ to $W$. Here

[TABLE]

denotes the pairing map that defines the dual vector space, and we denote elements of $V$ by $|q\rangle\in V$ and those of the dual vector space by $\langle v|\in V^{\ast}$, just so as to bring out the pattern better. Note that this pairing map is itself $k$-linear. Hence if we regard the ground field $k$ as the canonical 1-dimensional $k$-vector $\mathbf{1}$, as indicated, then this is actually a morphism in $\mathrm{Vect}_{k}^{\mathrm{fin}}$. There is also a closely related linear going the other way around:

[TABLE]

which, under the above identification, sends any element $c\in k$ to the linear map from $V$ to itself that is given by multiplication with $c$. One readily checks that these two functions make the following triangles commute

[TABLE]

whence called the triangle identities.

The quickest way to convince oneself that this indeed holds is to choose linear identifications $V\simeq\mathbb{R}^{n}$ and $W\simeq\mathbb{R}^{m}$, which means to choose linear bases. This in turn induces a canonical identification $V^{\ast}\simeq(\mathbb{R}^{n})^{\ast}\simeq\mathbb{R}^{n}$ (the dual linear basis), hence a linear identification

[TABLE]

of the vector space of linear maps $V\to W$ with the vector space of $n\times m$ matrices.

The description of dual vector spaces in terms of pairing and co-pairing maps satisfying triangle identities, as above, turns out to be equivalent to the traditional definition. It may seem more involved than the direct definition, but it has the great advantage that it makes sense without any actual reference to the nature of vector spaces: all that is needed to speak of dual objects is the analogue of the tensor product $\otimes$.

Categorical algebra shows that the triangle identities guarantee that $V^{\ast}\otimes W$ behaves like an “internalized” version of the Hom-set. The same applies to the tensor product of representations used in section 2.3.

Acknowledgements.* We thank Tim Dokchitser, James Dolan, James Montaldi and Todd Trimble for discussion. Our algorithm is inspired by the note [Tr09], which in turn goes back to private communication with James Dolan.*