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Hello *,

In tensor calculus (as used in general relativity and in other areas of 
physics) tensors with many upper and lower indices are often used. Their 
order is important even if some indices are upper and some lower: a tensor 
can have, say, the first index upper, the second one lower, the third 
upper, etc. As an exception, if a tensor is symmetric in a pair of 
adjacent indices, and one of them is upper and the other one is lower, 
they are traditionally written with the indices on top of each other (a 
typical example is the Kronecker \delta_\mu^\nu - it is easy to write in 
TeXmacs).

So, i've written the following the following scheme function:

(tm-define (tensor . args)
    (define (process-args l u d)
      (if (null? l)
          `(concat (rsup ,(apply tmconcat (reverse u)))
                   (rsub ,(apply tmconcat (reverse d))))
          (let ((x (tree->stree (car l))))
            (cond ((== (car x) 'rsup)
                   (process-args (cdr l) (cons (cadr x) u) (cons `(hphantom 
,(cadr x)) d)))
                  ((== (car x) 'rsub)
                   (process-args (cdr l) (cons `(hphantom ,(cadr x)) u) (cons 
(cadr x) d)))
                  ((== (car x) 'concat)
                   (let ((a (cadr x)) (b (caddr x)))
                     (cond ((== (car a) 'rsup)
                            (if (== (car b) 'rsub)
                                (process-args (cdr l) (cons (cadr a) u) (cons 
(cadr b) d))
                                (process-args (cdr l) (cons (cadr a) u) (cons 
`(hphantom ,(cadr a)) d))))
                           ((== (car a) 'rsub)
                            (if (== (car b) 'rsup)
                                (process-args (cdr l) (cons (cadr b) u) (cons 
(cadr a) d))
                                (process-args (cdr l) (cons (`(hphantom 
,(cadr a))) u) (cons (cadr a) d))))
                           (else (process-args (cdr l) u d)))))
                  (else (process-args (cdr l) u d))))))
    (tm->tree (process-args args '() '())))

If you put it to your ~/.TeXmacs/progs/my-init-texmacs.scm and start 
TeXmacs, you can do (in math mode, or course), e.g.,

R \extern <tab> tensor <tab> <tab> ^ m <tab> <right> <tab> _ n <tab> 
<right> <tab> ^ a <tab> <right> <tab> _ b <tab> <right> <enter>

and obtain a nice Riemann curvature tensor (with some particular positions 
of its 4 indices).

Several things should be done before this can become a perfect (or at 
least good) solution of the problem of writing tensor expressions in 
TeXmacs.

1. How to call it from the editor without writing \extern and other long 
stuff? From the documentation, xmacro has an arbitrary number of 
arguments; but I failed to connect some xmacro to extern scheme function 
tensor. If I do in the preamble

(assign "tensor" (xmacro "args" (extern "tensor" (arg "args"))))

(in scheme form), and then do

\tensor <enter> <tab> <some indices> <enter>

TeXmacs segfaults. So, at the moment I don't see how to assign some 
keyboard shortcut to this function.

2. Error handling in this function is poor. It takes any number of 
arguments. Each argument should have one of 3 forms:

^ <upper index>
_ <lower index>
^ <upper index> _ <lower index> (in any order)

If an argument contains something different, the function tensor can 
misbehave (some forms of garbage in arguments are ignored by the current 
code, but I have not checked this extensively).

3. ^ <upper index> _ <lower index> means that these indices are on top of 
each other. But indices can have different widths (an index may be 
\mu_{100), for example); then they are currently typeset very poorly. 
What's really needed is following. Suppose some argument of the function 
tensor is ^a_b. Then both a and b should be typeset in the width which is 
the maximum of the width of a and that of b. At the moment I don't know 
how to do this.

4. Currently indices are not editable in place. You have to press 
<backspace> after them, and then edit them is the passive form. It would 
be nice to be able to replace, say, mu by nu directly in the typeset 
tensor.

The problem 3 is not very important in practice: widely used symmetric 
tensors (like \delta, the Ricci tensor, the energy-momentum tensor) have 2 
indices, and they can be typeset in the standard TeXmacs way without using 
the function tensor. Theoretically, there can be a tensor with 10 indices 
symmetric in the 5-th and 6-th index, and I want to have 5--th upper and 
6-th lower, then the function tensor should be used and currently can 
produce poor results. But such cases seem rather exotic.

I'll be grateful for suggestions of improvements,

Andrey