Extremal word characters detect normalized cyclotomic simple labels

Let M be a nonzero finite-dimensional standard homogeneous cyclotomic module. Choose a nonzero word by successively maximizing the last constant-color block, then the preceding block after restricting that maximum, until the vacuum is reached. Write the resulting word as i_1^{r_1}...i_s^{r_s}, adjacent colors distinct. Restriction to a word idempotent is exact. Every simple constituent of M with nonzero word space must have precisely the same first maximal last-block length: a shorter length gives zero, a longer length would contradict maximality for M. The actual maximal-tail supplier identifies the full maximal restriction of this simple with N tensor L(i_s^{r_s}), where N is simple with zero residual i_s-tail. Repeat on N and on the corresponding exact restriction of M. At the end N is the unique vacuum simple. Reconstruct using the actual unique-head induction supplier. Hence exactly one simple shift orbit among the constituents of M can contribute at the chosen word. Its crystal label is obtained by the same successive right-induction powers from the vacuum, so is field independent under the native11 canonical B(Lambda) identification.

For the normalized simple S in that orbit, maximal restriction is normalized N tensor normalized L(i^r): restriction commutes with finite vector duality, the two factors are absolutely simple, and their unique self-dual normalizations force the remaining shift to be zero. The normalized rank-r nil-Hecke simple has graded dimension [r]i!, by the actual polynomial/corner calculation with shift -d_i binomial(r,2). Iterating gives gdim e(word)S=product_a [r_a]{i_a}!. Therefore the coefficient of [S] in [M] is gdim e(word)M divided by this nonzero product.

For character injectivity, realize the finite set of simple orbits appearing in a virtual relation as their direct sum, choose an extremal word as above, and evaluate the relation: exactly one orbit contributes, with its nonzero factorial polynomial. Its Laurent coefficient is zero. Remove it and repeat on the remaining finite set. This proves injectivity without assuming the relation is a module.