character_table_reps.at

the unpermuted versions are in cells.at

computing cell actions in presence of a permutation of the simple roots

embeddings of Weyl groups, not needed for induction, or at all really

just |reflection(L,i)| if |coroot(L,alpha)=coroot(G,alpha)|, as should be

equivalently |convert_to(G,W_elt(L,w)).matrix|, if coroots match as above

induction from a Levi factor

$L$ is subgroup of $G$, or more generally a group mapped homomorphically by
$f:L\to G$, and |pi_L| is a tabulated class function on W(L),
we want to compute the class function $pi_G=\ind_L^G(pi_L)$ on $W(G)$

formula: if C is a conjugacy class for G
    pi_G(C) = |W(G)| / (|W(L)|*|C|)) * \sum |C_j| pi_L(C_j)
where the sum runs over the W(L)-conjugacy classes C_j that constitute
$C\cap W(L)$, or more generally the inverse image $f^{-1}(C)$ (this follows
from Frobenius reciprocity, by which pairing $\ind_L^G(pi_L)$ with the
indicator $\chi$ function of $C$ equals pairing $pi_L$ with $\chi\circ f$.

In other words the value of |pi_L| on each class C_j in W(L) contributes to
the induced character value only at the class of f(C_j) in W(G), and for
this contribution its character value is multiplied by [W(G):W(L)] * |C|/|C_j|

algorithm: initialize the result to be the 0-character
run over conjugacy class representatives in L
for each conjugacy class representative w_L compute  w_G = class_of (w_L,G)
and add to the character value at (the class) w_G the  value:
pi_L(w_L) * index(W(G):W(L))*|conjugacy class of w_L|/|conjugacy class of w|
The factor by which pi_L(w_L) is multiplied is integer, as it is the index of
the centraliser subgroup of w_L in W_L inside its centraliser subgroup of w_G

P is a matrix taking simple roots of L=Wct_L.root_datum to the simple roots
 of a subgroup of G=Wct_G.root_datum
Note that P is not necessarily the matrix of an element of G
example:
L,M are Levi subgroups of G
L is G-conjugate to a Levi subgroup L' of M
we are given the character table of L, and want to induce W(L') -> W(M).
In this case P is the matrix of a Weyl group element of G (but not necessarily M)

the Steinberg character, for testing only

alternating sum, over all standard Levis, of induced from trivial;
should equal the sign character (if it does not, this function fails)