From 1622c55a1ad31d949140c9ed2cb689242f4d173e Mon Sep 17 00:00:00 2001 From: Hearot Date: Mon, 9 Oct 2023 16:15:01 +0200 Subject: [PATCH] feat(algebra1): aggiunge formula esplicita per sgn --- .../10. Il gruppo delle permutazioni/main.pdf | Bin 197549 -> 210749 bytes .../10. Il gruppo delle permutazioni/main.tex | 12 ++++++++++-- 2 files changed, 10 insertions(+), 2 deletions(-) diff --git a/Secondo anno/Algebra 1/10. Il gruppo delle permutazioni/main.pdf b/Secondo anno/Algebra 1/10. 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Il gruppo delle permutazioni/main.tex b/Secondo anno/Algebra 1/10. Il gruppo delle permutazioni/main.tex index 0711933..b789f44 100644 --- a/Secondo anno/Algebra 1/10. Il gruppo delle permutazioni/main.tex +++ b/Secondo anno/Algebra 1/10. Il gruppo delle permutazioni/main.tex @@ -59,8 +59,16 @@ In particolare, se $\sigma_k$ è un $k$-ciclo, $\sgn(\sigma_k) = (-1)^{k-1}$ e $\ord(\sigma_k) = k$. Si osserva inoltre che vi sono esattamente $\binom{n}{k} \frac{k!}{k} = \binom{n}{k} (k-1)!$ $k$-cicli in $S_n$ e che in generale l'ordine di una permutazione è il minimo comune multiplo degli - ordini dei suoi cicli. \medskip - + ordini dei suoi cicli. In particolare vale la seguente identità\footnote{ + Si verifica facilmente che il prodotto a destra fornisce un omomorfismo. Allora + è sufficiente mostrare che è ben definito e che vale $-1$ sulle trasposizioni. + Se si considera $\sigma = (a, b)$, per $i$ e $j$ tali per cui + $\{i, j\} \cap \{a, b\} = \emptyset$ il termine della produttoria è unitario; + per $\{i, j\} = \{a, b\}$ il termine è $-1$ e per un'intersezione di un solo + termine si osserva che vi sono due termini del prodotto che valgono $-1$ e + che moltiplicati si annullano nell'unità. Poiché $\sgn$ vale anch'esso $-1$ sulle trasposizioni, i due omomorfismi coincidono (infatti le trasposizioni generano $S_n$). + }: + \[ \sgn(\sigma) = \prod_{1 \leq i < j \leq n} \frac{\sigma(i) - \sigma(j)}{i - j}. \] Si definisce \textit{tipo} di una permutazione $\sigma$ la sua decomposizione in cicli disgiunti a meno degli elementi presenti nei cicli. Sia $\sigma$