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Francesco Polizzi
Francesco Polizzi
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Group with Finite group such that $K_{-1} \mathbb(\mathbb Z G = \mathbb Z / 2$)$ has non-trivial torsion

According to Carters Lower K-theory of finite groups for a finite group $G$ we have $$ K_{-1} \mathbb Z G = \mathbb Z^r \oplus \mathbb Z/2^s $$$$ K_{-1} (\mathbb Z G) = \mathbb Z^r \oplus \mathbb Z_2^s $$ where $s$ is the sum over all irreducible representations over $\mathbb Q$ which have even Schur index, but odd Schur index over $\mathbb Q_p$ for every finite prime which divides the order of $G$.

The integer $s$ seems to be $0$ for a pretty wide class of groups and its hard for me to even find a single concrete example of a finite group with non-trivial $s$. Does anyone know of a result in that direction?

Group with $K_{-1} \mathbb Z G = \mathbb Z / 2$

According to Carters Lower K-theory of finite groups for a finite group $G$ we have $$ K_{-1} \mathbb Z G = \mathbb Z^r \oplus \mathbb Z/2^s $$ where $s$ is the sum over all irreducible representations over $\mathbb Q$ which have even Schur index, but odd Schur index over $\mathbb Q_p$ for every finite prime which divides the order of $G$.

$s$ seems to be $0$ for a pretty wide class of groups and its hard for me to even find a single concrete example of a finite group with non-trivial $s$. Does anyone know of a result in that direction?

Finite group such that $K_{-1} (\mathbb Z G)$ has non-trivial torsion

According to Carters Lower K-theory of finite groups for a finite group $G$ we have $$ K_{-1} (\mathbb Z G) = \mathbb Z^r \oplus \mathbb Z_2^s $$ where $s$ is the sum over all irreducible representations over $\mathbb Q$ which have even Schur index, but odd Schur index over $\mathbb Q_p$ for every finite prime which divides the order of $G$.

The integer $s$ seems to be $0$ for a pretty wide class of groups and its hard for me to even find a single concrete example of a finite group with non-trivial $s$. Does anyone know of a result in that direction?

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Georg Lehner
Georg Lehner
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Group with $K_{-1} \mathbb Z G = \mathbb Z / 2$

According to Carters Lower K-theory of finite groups for a finite group $G$ we have $$ K_{-1} \mathbb Z G = \mathbb Z^r \oplus \mathbb Z/2^s $$ where $s$ is the sum over all irreducible representations over $\mathbb Q$ which have even Schur index, but odd Schur index over $\mathbb Q_p$ for every finite prime which divides the order of $G$.

$s$ seems to be $0$ for a pretty wide class of groups and its hard for me to even find a single concrete example of a finite group with non-trivial $s$. Does anyone know of a result in that direction?