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Let $\Lambda$ be an integral lattice with the symmetric bilinear form (or the pairing) $\langle,\rangle$.

I'd like to know any reference (or even just the standard math terminology) for (the properties of) homomorphisms (as an Abelian group) $\sigma:\Lambda\to\Lambda$ such that

  • It reverses the pairing $\langle \sigma a,\sigma b\rangle=-\langle a,b\rangle$
  • The homomorphism $\sigma_D$ induced on $D=\Lambda^*/\Lambda$ is an involution, $\sigma_D{}^2=1$.

(I'm not sure if it's really a math-research level question; but it's related to my physics research.)

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  • $\begingroup$ The info on this post mathoverflow.net/questions/58825/… almost fixes the structure of $D$ and $\sigma_D$ if $\langle a,\sigma_D a\rangle=0$ for $a\in D$; but in my case $\langle a,\sigma_D a\rangle$ can be $1/2 \mod \mathbb{Z}$. $\endgroup$
    – Yuji Tachikawa
    Commented Aug 7, 2016 at 6:52
  • $\begingroup$ The case with $\langle a,\sigma_D a\rangle=1/2$ was covered in CTC Wall, "QUADRATIC FORMS ON FINITE GROUPS, AND RELATED TOPICS ". $\endgroup$
    – Yuji Tachikawa
    Commented Aug 7, 2016 at 13:28

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