Definition of the Gauss symbol [closed]

Question

Tahara refers to the "Gauss symbol" in the article, On the second cohomology groups of semidirect products, Math. Z. 129 (1972) 365--379. For a fixed $n$, let $S_{ij}$ be the expression \begin{equation} \left[ \frac{i+j}{n}\right] -\left[ \frac{i}{n}\right] - \left[ \frac{j}{n}\right] \end{equation} which comes up often in the paper. What does $\left[ \text{ }\right] $ mean here? Also, what then are $S_{k,-k}$ and $S_{k,k}$, assuming $k$ and $n$ are coprime?

Answer 1

Gerry and Emil in the comments are correct: it's the integer floor function.

Mastnak (citation below) considers Tahara's work in an appendix, "An excursion into group cohomology." On p433 in Example A.6, working in the order $n$ cyclic group generated by $x$, Mastnak uses

...the cocycle defined by $c_a(x^i, x^j) = a^{\lfloor \frac{i+j}{n}\rfloor}$ for $i,j \in \{0,1,\ldots,n-1\}$.

This is followed by

Remark. We can drop the condition $i,j \in \{0,1,\ldots,n-1\}$ in definition of $c_a$ if we replace $\lfloor \frac{i+j}{n}\rfloor$ by $\lfloor \frac{i+j}{n}\rfloor - \lfloor \frac{i}{n}\rfloor - \lfloor \frac{j}{n}\rfloor$.

For $i=k, j=-k$, suppose $k = an+b > 0$ with $a \in \Bbb{Z}$ and $b \in \{1,\ldots,n-1\}$ (if $b=0$ then $n|k$ contrary to your coprime condition, and the case $n=1$ doesn't make much sense in this context). Then \begin{equation} \left\lfloor \frac{0}{n}\right\rfloor - \left\lfloor \frac{k}{n}\right\rfloor - \left\lfloor \frac{-k}{n}\right\rfloor = 0 - a - (-a-1) = 1.\end{equation}

For the $i=k, j=k$ question added later, use the same set up. Then \begin{equation} \left\lfloor \frac{2k}{n}\right\rfloor - 2\left\lfloor \frac{k}{n}\right\rfloor = \left\lfloor \frac{2an+2b}{n}\right\rfloor - 2a = \begin{cases} 2a-2a = 0 & \text{if } 2b < a, \\ 2a+1 - 2a = 1 & \text{if } 2b \ge a. \end{cases}\end{equation}

As to Tahara calling it the Gauss symbol, here's a passage from Wolfram MathWorld on the floor function and its evolving notation.

Unfortunately, in many older and current works (e.g., Honsberger 1976, p. 30; Steinhaus 1999, p. 300; Shanks 1993; Ribenboim 1996; Hilbert and Cohn-Vossen 1999, p. 38; Hardy 1999, p. 18), the symbol $[x]$ is used instead of $\lfloor x \rfloor$ (Graham et al. 1994, p. 67). In fact, this notation harks back to Gauss in his third proof of quadratic reciprocity in 1808. However, because of the elegant symmetry of the floor function and ceiling function symbols $\left\lfloor x \right\rfloor$ and $\lceil x \rceil$, and because $[x]$ is such a useful symbol when interpreted as an Iverson bracket, the use of $[x]$ to denote the floor function should be deprecated.

Mitja Mastnak (2002), Hopf algebra extensions arising from semi-direct products of groups, J. Alg. 251: 413--434 (MSN).