Expressive macro for tensors; raised and lowered indices

Question

Does anyone have a satisfying solution to the problem of typesetting tensors with raised/lowered indices? For example, I can write the following equation:
ddot x^mu = Gamma^{mu}{}_{alpha}{}_{beta} dot x^alpha dot x^beta

When writing a lot of tensors, this is cumbersome.
What I'm looking for is a way to construct a command which can produce tensors like Gamma above with more expressive syntax. For example, a command tens that operates like this would be ideal:
ddot tens{x}{mu} = tens{Gamma}{mu}[alpha][beta] dot tens{x}{alpha} tens{x}{beta}
The key quality of my desired syntax is that there is an arbitrary number of arguments of two distinct types. Argument encased in { } are raised indices, while arguments encased in [ ] are lowered indices. I don't need a command that looks exactly like this; I am looking for something that is similarly expressive.
Does anyone have a solution to this problem? An implementation of a tens command that works as above? I don't know how to do this.

egreg · Accepted Answer

In my opinion, the subscripts and superscripts are a single argument.
You can use the tensor package, without reinventing the wheel: it has a very handy syntax.
I also provide a tens command according to your preferences.
documentclass{article}
usepackage{tensor}

%usepackage{xparse}

ExplSyntaxOn
NewDocumentCommand{tens}{mo}
 {
  #1
  IfNoValueTF { #2 } 
   {
    __myridium_tens_up_lookup:
   }
   {
    __myridium_tens_down_lookup: [ #2 ]
   }
 }

cs_new_protected:Nn __myridium_tens_down_lookup:
 {
  peek_charcode_ignore_spaces:NTF [
   {
    __myridium_tens_down:w
   }
   { kern2scriptspace }
 }
cs_new_protected:Npn __myridium_tens_down:w [ #1 ]
 {
  {mathstrut}
  sb{#1}
  kern-scriptspace
  __myridium_tens_up_lookup:
 }
cs_new_protected:Nn __myridium_tens_up_lookup:
 {
  peek_catcode_ignore_spaces:NTF c_group_begin_token
   {
    __myridium_tens_up:n
   }
   { kern2scriptspace }
 }
cs_new_protected:Nn __myridium_tens_up:n
 {
  {mathstrut}
  sp{#1}
  kern-scriptspace
  __myridium_tens_down_lookup:
 }
ExplSyntaxOff

begin{document}

subsection*{With texttt{tensor}}
[
tensor{ddot{x}}{^mu}=
tensor{Gamma}{^mu_alpha_beta}
tensor{dot{x}}{^alpha} tensor{dot{x}}{^beta}
]

[
tensor{Gamma}{_mu^nu^rho_alpha^nu^rho}
tensor{dot{Gamma}}{_mu^nu^rho_alpha^nu^rho}
]

subsection*{With the hand-made macro}
[
tens{ddot{x}}{mu}=
tens{Gamma}{mu}[alphabeta]
tens{dot{x}}{alpha} tens{dot{x}}{beta}
]

[
tens{Gamma}[mu]{nurho}[alpha]{nurho}
tens{dot{Gamma}}[mu]{nurho}[alpha]{nurho}
]

end{document}

user229669 · Answer

Here is something that works like you describe but with round brackets instead of curly ones. As usual such things can be a bit fragile, so occasionally you need to relax a bit to mark it fully work, as can be seen in the second example.
documentclass{article}
makeatletter
edeftens@u{(}
edeftens@l{[}
deftens@U#1)#2{{}^{#1}expandaftertens@i#2relax}
deftens@L#1]#2{{}_{#1}expandaftertens@i#2relax}
deftens@i#1#2{edeftens@t{#1}%
ifxtens@ttens@u
expandaftertens@U#2
else
ifxtens@ttens@l
expandaftertens@L#2
else
#1#2
fi
fi}
deftens#1#2{#1expandaftertens@i#2}
makeatother
begin{document}
begin{tabular}{rl}
works: &
$tens{Gamma}[mu](nurho)[alpha](nurho) dottens{x}(alpha)
dottens{x}(beta)$ [2em]

does not work: & 
$tens{Gamma}[mu](nurho)[alpha](nurho) dottens{x}(alpha)
tens{x}(beta)$ [2em]

relax and it works again: &
$tens{Gamma}[mu](nurho)[alpha](nurho) dottens{x}(alpha)relax
tens{x}(beta)$ 
end{tabular}
end{document}

To be clear: such macros are mainly for recreation purposes and not for the real world. These days the LaTeX world has enough other problems...

Gaussler · Answer

I wouldn’t use such a syntax, but SemanTeX can be set up to accomplish something resembling this (disclaimer: I am the author). Note that you will need a recent update of SemanTeX (October or later, I think) for this example to work. Note that I also prefer defining keys dot and ddot instead of directly using the commands dot and ddot.
documentclass{article}

usepackage{semantex}

NewVariableClasstens[
    output=tens,
    definekeys={
        {dot}{ command=dot },
        {ddot}{ command=ddot },
        {preindex}{ rightreturn, symbolputright={{}} },
        {postindex}{ rightreturn, symbolputright=kern-scriptspace },
    },
    definekeys[1]={
        {default}{ preindex, lower={#1}, postindex },
        {arg}{ preindex, upper={#1}, postindex },
    },
]

begin{document}

$ tens{dot x}{mu} = tens{dot{Gamma}}{mu}[alpha][beta] 
    tens{dot{x}}{alpha} tens{dot{x}}{beta} $

$ tens{ddot x}{mu} = tens{dot{Gamma}}{mu}[alpha][beta]
    tens{dot{x}}{alpha} tens{dot{x}}{beta} $

$ tens{x}[ddot]{mu} = tens{Gamma}[dot]{mu}[alpha][beta]
    tens{x}[dot]{alpha} tens{x}[dot]{beta} $

end{document}

Personally, I would prefer to use a more keyval-based syntax, as below:
documentclass{article}

usepackage{semantex}

NewVariableClassTensor[
    output=Tensor,
    definekeys={
        {dot}{ command=dot },
        {ddot}{ command=ddot },
        {preindex}{ rightreturn, symbolputright={{}} },
        {postindex}{ rightreturn, symbolputright=kern-scriptspace },
    },
    definekeys[1]={
        {up}{ preindex, upper={#1}, postindex },
        {low}{ preindex, lower={#1}, postindex },
    },
]

begin{document}

$ Tensor{x}[dot,up=mu] = Tensor{Gamma}[dot,up=mu,low=alpha,low=beta]
    Tensor{x}[dot,up=alpha] Tensor{x}[dot,up=beta] $

NewObjectTensortGamma{Gamma}
NewObjectTensortx{x}

$ tx[dot,up=mu] = tGamma[dot,up=mu,low=alpha,low=beta]
    tx[dot,up=alpha] tx[dot,up=beta] $

end{document}

wipet · Answer

The definition of your desired macro tens using TeX primitives follows:
deftens#1{#1futureletnexttensA}
deftensA{deftensX{}%
   ifxnext[deftensX[##1]{{}_{##1}futureletnexttensA}fi 
   ifxnextbgroup deftensX##1{{}^{##1}futureletnexttensA}fi
   tensX}

%% test:

$tensGamma [mu]{nurho}[alpha]{nurho}$

$ddottens{x}{mu} = tens{Gamma}{mu}[alpha][beta] 
                      dottens{x}{alpha} dottens{x}{beta}$

Expressive macro for tensors; raised and lowered indices

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