fix: proofread Representation Theory part

[vEnhance]

Proofreading pass over the Representation Theory part (issue #315), covering tex/rep-theory/rep-alg.tex, semisimple.tex, characters.tex, and applications.tex. All findings are minor (typos, grammar, notation). No mathematical arguments were rewritten.

semisimple.tex

• Transposed matrix dimensions. Napkin's sec:vector_space_linear_maps fixes the convention that a map from an $m$-dimensional domain to an $n$-dimensional codomain is an $n \times m$ matrix (codomain rows × domain columns). Two spots had the order reversed:
  • "$\Homrep(V^{\oplus n}, V^{\oplus m})$ consist of $n \times m$ matrices" → consists of $m \times n$. The map is $V^{\oplus n}\to V^{\oplus m}$, and the displayed matrix immediately after is genuinely $m$ rows × $n$ columns.
  • "a linear map from an $n$-dimensional ... to an $m$-dimensional ... can be written as a $n \times m$ matrix" → $m \times n$, matching the cited section.
• "the second arrow is inclusion to the $j$th component" → projection onto (the arrow in the diagram is the surjection $V^{\oplus n}\surjto V$).
• Stray matrix index in the "stacked horizontally" remark: row 2 had c_{(m-1)1} → c_{(m-1)2}.
• "sends $e_1$ to $w_1$ to $e_2$ to $w_2$" → "and $e_2$".
• "to breaks any finite-dimensional representation" → "to break".
• \bigoplus_I → \bigoplus_i (capital subscript typo).
• Grammar: "a linear maps" → "a linear map"; "irreps ... doesn't talk" → "don't talk".

characters.tex

• "the character $\chi_V$ ... attached to $A$" → attached to $V$ (the character is attached to the representation).
• "characters of finite-dimensional irreps of $V$" → "irreps of $A$".
• Notation: "$A = \Mat_k(d)$" → "$A = \Mat_d(k)$" (matches usage everywhere else).

applications.tex

• "$\Tr T = \lambda_i \dim V$" → "$\Tr T_i$" (the operator is $T_i$).
• "be an complex irrep" → "a complex irrep".
• "second orthogonality formula applied $g$ and $1_G$" → "applied to $g$ and $1_G$".
• "each element of $H$ acts trivially in $G$" → "acts trivially on $V$" (the point is that $H$ lies in $\ker\rho$, so it acts as the identity on $V$).

No issues found in rep-alg.tex.

Part of the proofreading campaign (#315).

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https://claude.ai/code/session_01W4ky3kJnfS3bwr4ZZrvbKk

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