Exercise PLAN-108

A finite-dimensional trace character is a conjugacy-invariant K-valued function with chi(e)=(dim_K V)*1_K. Integer degree is uniquely recovered in characteristic zero, but generally only its residue is visible in positive characteristic.

1. Multiplicativity of phi and preservation of identity/inverses express the conjugated operator as phi(h)phi(g)phi(h)^(-1) over any field.
2. Cyclicity of the finite-dimensional K-valued trace proves similarity and conjugacy invariance without unitarity or a finite-group hypothesis.
3. Conjugacy invariance makes chi_bar([g])=chi(g) well-defined and uniquely factors chi through the set of conjugacy classes.
4. The identity trace is n*1_K, including n=0; scalar typing, characteristic-zero injectivity and explicit characteristic-p examples distinguish this identity from integer-degree recovery.

The trace takes values in K, while n=dim_K(V) is a nonnegative integer. Let iota:N->K be iota(n)=n*1_K. Writing these types explicitly avoids either changing the field or mistaking a field scalar for an integer.

A representation is a homomorphism into the group of invertible K-linear maps. In particular phi(e)=I_V (cancel phi(g) from phi(e)phi(g)=phi(g)), and phi(h^(-1))=phi(h)^(-1) because their products in both orders equal phi(e). Consequently phi(hgh^(-1))=phi(h)phi(g)phi(h)^(-1). For n-by-n matrices A,B over the COMMUTATIVE field K, tr(AB)=sum_ij A_ij B_ji=sum_ji B_ji A_ij=tr(BA). Applying this with A=phi(h) and B=phi(g)phi(h)^(-1), or moving the last factor cyclically to the front, gives chi(hgh^(-1))=tr(phi(h)^(-1)phi(h)phi(g))=tr(phi(g))=chi(g). The same calculation shows trace is invariant under change of basis, so chi is well-defined independently of the matrix representative. No inner product, finite-group averaging or unitary structure is used. Linearity is a trace property, but no linear combination of representation matrices is needed for this conjugation computation; the actual ingredients are cyclicity, multiplicativity and preservation of identity/inverses.

For the quotient SET of conjugacy classes, define chi_bar([g])=chi(g). If g' is conjugate to g, the calculation just proved says chi(g')=chi(g), so the definition does not depend on the representative. Equivalently chi=chi_bar composed with the quotient map, and surjectivity of that quotient map gives uniqueness of chi_bar. No finiteness of the set of classes is needed, and no identification with the center of a group algebra is asserted for arbitrary infinite G.

At the identity, an n-element basis gives the identity matrix with n diagonal entries 1_K. Hence chi(e)=tr(I_V)=sum_(j=1)^n 1_K=iota(n), universally over K. Thus the old notation chi(e)=dim V is valid shorthand IF its right side explicitly means this scalar image. As an untyped equality with an integer, or the answer's assertion of universal integer-degree recovery, it is not a justified field-independent claim. In characteristic zero the natural map is injective: (n-m)*1_K=0 forces n=m for nonnegative integers n,m. Accordingly knowing the scalar iota(n) determines n uniquely; over C one ordinarily identifies that scalar with the usual integer degree. This is uniqueness, not an asserted efficient extraction algorithm from an arbitrary field-element encoding.

In characteristic p>0, iota(n) depends only on n modulo p. For the trivial group and its trivial representation on V=F_p^p, phi(e)=I_p and chi(e)=p*1_(F_p)=0, but dim(V)=p, not zero as an integer. More strongly, the trivial representations on F_p and F_p^(p+1) have the same entire trace character (the constant scalar 1) yet different integer dimensions. This counterexample does not refute chi(e)=iota(n); it refutes integer recovery without a characteristic-zero assumption or an independent dimension restriction. Even if n<p happens to hold, that is extra information, not a premise of the old question. For n=0, the zero vector space has its unique invertible endomorphism as identity, the empty trace sum is zero, and chi is identically zero; all homomorphism, conjugacy and quotient arguments remain valid. A zero-dimensional trace character can coincide with a positive-dimensional one in characteristic p. The correction therefore preserves the original class-function/identity-trace family while explicitly repairing scalar typing and the claimed conclusion, rather than narrowing all representations to the complex case.

1. Proves identity/inverse preservation and the representation conjugation expression over arbitrary K without adding finite-G or unitarity assumptions. (1 weight)
2. Establishes trace cyclicity and similarity invariance with field-valued typing; identifies the actual proof ingredients, not a spurious use of trace linearity. (1 weight)
3. Defines and proves the well-defined unique factorization through conjugacy classes. (1 weight)
4. Proves chi(e)=n*1_K; assesses rather than silently recasts the old degree assertion, handles characteristic zero and positive characteristic with explicit unequal-dimension examples, and treats n=0. (1 weight)

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