A Topological Proof for a Version of Artin's Induction Theorem

We define a Euler characteristic $\chi(X,G)$ for a finite cell complex $X$ with a finite group $G$ acting cellularly on it. Then, each $K_{i}(X)$ (a complex vector space with basis the $i$-cells of $X$) is a representation of $G$, and we define $\chi(X,G)$ to be the alternating sum of the representations $K_{i}(X)$, as elements of the representation ring $R(G)$ of $G$. By adapting the ordinary proof that the alternating sum of the dimensions of the chain complexes is equal to the alternating sum of the dimensions of the homology groups, we prove that there is another definition of $\chi(X,G)$ with the alternating sum of the representations $H_i(X)$, again as elements of the representation ring $R(G)$. We also show that the character of this virtual representation $\chi(X,G)$, with respect to a given element $g$, is just the ordinary Euler characteristic of the fixed-point set by this element. Finally, we give a topological proof of a version of Artin's induction theorem. More precisely, we show that, if $G$ is a group with an irreducible representation of dimension greater than 1, then each character of $G$ is a linear combination with rational coefficients of characters induced up from characters of proper subgroups of $G$.


Introduction
The origin of the representation theory of finite groups goes way back to a correspondence between R. Dedekind and F. G. Frobenius that took place around a century ago [2,1]. This theory studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. Specifically, a representation is a way of making an abstract algebraic object more solid by defining its elements by matrices and arithmetic operations on matrices. Linear algebra and matrix theory are well-known and less abstract, so understanding more abstract objects using familiar linear algebra objects can be helpful in deriving properties and simplifying calculations in more abstract settings. Representation theory is a branch of abstract mathematics with many important applications not only in mathematics but in other sciences as well, including chemistry, physics, computing and statistics. To name a few examples: in [3], an algorithm to construct skew-symmetric matrices is developed by using real irreducible representations of SO(3) (application in Lie theory); in [4], a non-Abelian 4 dimensional unitary representation of the Braid group is used to obtain a totally leakage-free braiding (application in computer science); and in [5], applications of the group representation theory in various branches of physics and quantum chemistry, in particular nuclear and molecular physics, are discussed. The motivation for this work was to use representation theory to give a topological proof for an abstract problem. We explain the method of research below: for X, a finite cell complex, let K * (X) be the cellular chain complex with complex coefficients where each of the K i (X)s is a complex vector space with the i-cells as a basis. Now, suppose a finite group G acts cellularly on X. Then, each K i (X) is a representation of G, and we can make a more accurate Euler characteristic for X with G acting by taking χ(X, G) = n i=0 (−1) i [K i (X; C)] ∈ R C (G). The dimension of this virtual representation is just the ordinary Euler characteristic of X. We claim there is another definition of χ(X, G) as follows, is the ith homology group of X with G acting on it. In this paper, we will work on the following problems. We will try to adapt the ordinary proof to give a similar formula with equality as elements of R C (G) when G acts on X. We will also look for a formula for the character of χ(X, G) in terms of the ordinary Euler characteristic of X g = {x ∈ X| gx = x}. Finally, with the help of these results, we give a topological proof of a version of Artin's induction theorem. Artin's induction theorem says that any character of a finite group can be expressed as a rational linear combination of characters that are induced from the cyclic subgroups. In this work, we prove a version of this theorem; more precisely, we show that if G is a group with an irreducible representation of dimension greater than 1, then each character of G can be expressed as a rational linear combination of characters that are induced up from characters of proper subgroups of G.
The structure of the paper is as follows. Section 2 will give the preliminaries on representation theory and topology. Sections 3 and 4 will give examples for the Euler characteristic computation using definition with the chain groups and the homology groups, respectively. Finally, we give the results in Section 5.
This work is presented in International Congress of Mathematicians, ICM2014, which took place in Seoul, South Korea in 2014.

Preliminaries
Our main references here are [6,7,8]. Note that the underlying field is the field of complex numbers.
Definition 2.1. Let ϕ : G → GL(V ) be a linear representation, and let W be a subspace of V . Suppose that W is invariant under the action of G, that is to say, suppose that w ∈ W implies that ϕ s (w) ∈ W for all s ∈ G. The restriction ϕ W of ϕ s to W is then an isomorphism of W onto itself. Thus, ϕ W : G → GL(W ) is a linear representation of G in W and is said to be a subrepresentation of G.

Definition 2.2. A non-zero linear representation is said to be irreducible if it has no proper non-trivial subrepresentation.
Definition 2.3. Let G be a finite group, let H be a subgroup of index n, and let (π, V ) be any representation of H. Let x 1 , x 2 , . . . , x n be representatives in G of the cosets in G/H. The induced representation Ind H G π acts on W = x i V , where i ranges over the coset representatives, and via this, G acts on W as follows: Below, we state some well-known theorems on representation theory that will be useful in the coming sections.
Theorem 2.1 (Fixed-Point Formula). Let V be a representation of a finite group G, and let X be a finite G-set. Then, the number of left fixed elements (by the action of g) in X is χ V (g) for every g ∈ G.
Theorem 2.2. The number of conjugacy classes of G is the same as the number of irreducible representations of G, up to isomorphism.  Theorem 2.5. Let V be a linear representation of G with character φ, and suppose V decomposes into a direct sum of irreducible representations V = W 1 ⊕ . . . ⊕ W k . Then, if W is an irreducible representation with character χ, the number of W i isomorphic to W is equal to the scalar product (φ|χ), where Next, we state some well-known theorems in algebraic topology. K below is a finite simplicial complex.
Theorem 2.6. H 0 (K) is a free Abelian group whose rank is the number of connected components of |K|.
Theorem 2.7. If |K| is connected, Abelianizing its fundamental group gives the first homology group of K.
Remark 2.1. Examples in Sections 3 and 4 below give evidence for Theorem 5.1, which will be proved in Section 5.

Evaluating χ(X, G) Using the Definition with the Chain Groups
In this section, we evaluate χ(X, G) for some examples using its definition with the chain groups.
Example 3.1. D 3 acting on the equilateral triangle as the group of symmetries.
D 3 here stands for the dihedral group of degree 3, or in other words, the dihedral group of order 6. It is the group of 6 symmetry transformations of an equilateral triangle. They are reflections in the axes through 3 vertices and also rotations around the center by multiples of 120 • . Let us denote the rotations by e, ρ, ρ 2 and the reflections by α, β, γ. The group D 3 has 2 one-dimensional (trivial and alternating), and 1 twodimensional (standard) irreducible representations; see [8,7]. Let us call them 1, σ, τ , respectively. For the trivial representation, χ 1 (g) = 1 for all g ∈ G. Character values of the alternating representation are given by χ σ (g) = 1 for g ∈ A 3 and χ σ (g) = −1 for g ∈ A 3 . Finally, the 2 dimensional τ has character values χ τ (e) = 2, χ τ (α) = 0 and χ τ ((123)) = −1.
Here, K 0 is the representation space of dimension 3, which is the number of 0-cells. Let P be the representation corresponding to this. By the fixed-point formula, Theorem 2.1, the character values of this representation are shown in Table 1. By Theorem 2.5, the number of irreducibles isomorphic to 1 is (χ, φ 1 ) = 1, where φ 1 stands for the character of the trivial representation. Similarly, (χ, φ 2 ) = 0 and (χ, φ 3 ) = 1, where φ 2 and φ 3 are the characters of σ and τ , respectively. Therefore, P = 1 ⊕ τ.
Similarly, K 1 is the representation space of dimension 3. Let us call this representation Q. The character values for Q are presented in Table 2.
C 2 here represents the cyclic group of order 2. Thus the Abelian group C 2 × C 2 has four elements, identity element of order 1, and the remaining three elements of order 2. Let us represent these elements by e, ρ 1 , ρ 2 , ρ 3 , where ρ 1 is the axis joining vertices 5 and 6, ρ 2 is the axis joining vertices 3 and 4, and finally ρ 3 is the axis joining vertices 1 and 2; see  As this group is Abelian, all the irreducible representations have a degree of 1. Furthermore, as the sum of the squares of dimensions of distinct irreducible representations is on the order of C 2 × C 2 , there must be 4 of them. By [8,7], these are shown in Table 4.
Let P be the representation corresponding to the space K 0 of dimension 6. Similar work to the previous example shows that P = 31 ⊕ T 1 ⊕ T 2 ⊕ T 3 . Next, let Q be the representation corresponding to the space K 1 of dimension 12. We obtain Q = 31 ⊕ 3T 1 ⊕ 3T 2 ⊕ 3T 3 . Finally, if R is the representation corresponding to the space K 2 , of dimension 8, then R = 21 ⊕ 2T 1 ⊕ 2T 2 ⊕ 2T 3 . Thus, we obtain Example 3.3. C 2 × C 2 × C 2 acting on the octahedron as reflections in the three coordinate planes.
Following from the previous example, C 2 ×C 2 ×C 2 is an Abelian group of order 8 with an identity element of order 1 and the remaining seven elements of order 2. Call the α reflection in the xy-plane, β reflection in the yz-plane, and γ reflection in the xz-plane; see Figure 2. Hence, the elements of C 2 × C 2 × C 2 are e, α, β, γ, α • β, α • γ, β • γ, α • β • γ (we will use the notation αβ for the element α • β). Similar calculations to previous examples show that χ(X, G) = V 1 ⊕ V 8 , where V 1 is the trivial representation, and V 8 is the one mapping e, αβ, αγ, βγ to 1 and the rest of the elements to −1. (The group C 2 × C 2 × C 2 is Abelian; therefore, it has 8 irreducible representations of degree 1; see Table 5 [8,7]).
x y z Figure 2: C 2 × C 2 × C 2 acting on the octahedron.
Evaluating the χ(X, G) Using the Definition with the Homology Groups In this section, we will evaluate χ(X, G) for the examples above, this time by using its definition with the homology groups.   The group H 0 is isomorphic to C contributing 1 to [H 0 ]. H 1 ∼ = 0, so there is no contribution to [H 1 ], and finally, H 2 ∼ = C with dimension 1. If P is the representation corresponding to this, then P has character values 1 and −1 for the rotations and the reflections, respectively. As stated earlier, this representation is V 8 . Hence, χ(X; G) = 1 ⊕ V 8 .

Results
Theorem 5.1. As elements of the representation ring, for a general case.
Proof. Let Z n be the group of n-cycles and B n be the group of bounding n-cycles. There are two short exact sequences 0 → Z n K n B n−1 → 0 0 → B n Z n H n → 0 and these sequences split. Thus, we have K n = Z n ⊕ B n−1 and Z n = B n ⊕ H n . This implies that K n = B n ⊕ H n ⊕ B n−1 . Therefore, χ Kn = χ Bn + χ Hn + χ Bn−1 .
By the above equation, we have All χ Bi s and χ Bi−1 s cancel except for χ B−1 and χ Bn , so let us consider these cases now. We have χ B−1 = 0 since χ K0 = χ B0 + χ H0 . Additionally, B n = 0, as there are no {n + 1}-dimensional faces. Hence, χ Bn = 0. Therefore, Proposition 5.1. The character of χ(X; G) is equal to the ordinary Euler characteristic of X g where X g = {x ∈ X| gx = x}.

Conclusions
In this paper, given a finite cell complex X with a finite group G acting cellularly on it, we define a more accurate Euler characteristic for the finite cell complex with this action as alternate sums of the chain complexes and the homology groups considered as elements of the representation ring. We prove that both of the definitions are equivalent and that the character of this virtual representation with respect to a given element g is just the ordinary Euler characteristic of the fixed point set by this element. Finally we use representation theory to give a topological proof of a weaker version of Artin's induction theorem. E. Artin's induction theorem says that any character of a finite group can be expressed as a rational linear combination of characters that are induced from the cyclic subgroups. In our work we prove a version of this theorem; in fact, we show that, if G is a group with an irreducible representation of dimension greater than 1, then each character of G is a rational linear combination of characters induced up from characters of proper subgroups of G.

Competing interests
The author declares no conflict of interest.

Funding
This research received no external funding.