# Interior product (contraction) symbol

See the above excerpt from John Lee's Introduction to Smooth Manifolds. In differential topology and linear algebra, there's an operator called the interior product or contraction operator, denoted $\iota_X(\omega)$ or alternatively with some other "half box" like operator $X_|\omega$. Does anyone know how to markup that operator in LaTeX?

Google has turned up nothing except for this thread: Symbol for contraction / interior product?

That thread did receive an accepted answer, which was \invneg from the MnSymbol package, but the suggestion given there does not seem correct to me. Or at least, it does not match what I'm familiar with. That answer results in an inverted negation symbol, whereas the contraction operator should have a stroke down at the floor of the line, and a long stroke going up almost the full height of the line. It can be found in many texts, but why can I not find the correct LaTeX markup?

• Dec 19, 2011 at 21:44
• Is the following link a good reference that shows the notation you're after: The Interior Product? We're after an image to see whether this could be contained within an existing font or command (and therefore perhaps be solved by @JosephWright's link), or perhaps requires a larger construct.
– Werner
Dec 19, 2011 at 21:45
• The rendering of the character in Asana Math and in the STIX fonts is just like a "reversed \lnot". Dec 19, 2011 at 21:50
• detexify doesn't recognize it Dec 20, 2011 at 10:46
• I'm reading a paper which uses \rfloor (it's on arxiv so you can read the source), but it looks terrible and nonstandard. Dec 23, 2011 at 19:27

I stumbled upon the correct markup used in Wikipedia's article on geometric algebra. I looked at the code and found the symbol is given by \lrcorner. \llcorner is the same thing but flipped horizontally.

• need: \usepackage{amssymb} Mar 17, 2021 at 2:47

You can import the symbol from MnSymbol by saying

\DeclareFontFamily{U}{MnSymbolC}{}
\DeclareSymbolFont{MnSyC}{U}{MnSymbolC}{m}{n}
\DeclareFontShape{U}{MnSymbolC}{m}{n}{
<-6>  MnSymbolC5
<6-7>  MnSymbolC6
<7-8>  MnSymbolC7
<8-9>  MnSymbolC8
<9-10> MnSymbolC9
<10-12> MnSymbolC10
<12->   MnSymbolC12}{}
\DeclareMathSymbol{\intprod}{\mathbin}{MnSyC}{'270}


This way you won't override your math fonts with MnSymbol.

The alternative way with reversing the \lnot symbol is

\usepackage{graphicx}
\newcommand{\intprod}{\mathbin{\raisebox{\depth}{\scalebox{1}[-1]{$\lnot$}}}}


If the symbol is needed also in other sizes, then

\newcommand{\intprod}{\mathbin{\mathpalette\dointprod\relax}}
\newcommand{\dointprod}[2]{%
\raisebox{\depth}{\scalebox{1}[-1]{$#1\lnot$}}}


Here's a picture of the output in the two ways.

Inspired in the answers by @egreg and @alex-eftimiades my solution was to define the \lrcorner symbol as a mathematical binary operator, i.e.

\newcommand{\iprod}{\mathbin{\lrcorner}}


So, putting it at work

  \begin{equation*}
\begin{split}
X \iprod (\omega_1 \we \omega_2) & = (X \iprod \omega_1) \we \omega_2 + (-1)^r \omega_1 \we (X \iprod \omega_2)
\\
X(\omega) & = X \iprod \de{\omega} + \de{(X \iprod \omega)}
\\
X(Y \iprod \omega) & = [X,Y] \iprod \omega + Y \iprod X(\omega)
\end{split}
\end{equation*}


yields

• This is exactly what I needed, thank you very much! Dec 16, 2020 at 10:22