CHEVIE
web GAP3 manual
web CHEVIE homepage
txt Installation instructions on Linux
Contents:
- Characters and Conjugacy Classes
- Generalizing Haiman’s Conjecture
- Scripts
1. Characters and Conjugacy Classes
CHEVIE Data
pdf Poster
| \(\mathtt{CharInfo}\) | \(\mathtt{ICCTable}\) | \(\mathtt{X}\) | \(\mathtt{Y}\) | \(\mathtt{byFamily}\) | |
|---|---|---|---|---|---|
| \(B_2\) | txt | txt | txt | txt | txt |
| \(G_2\) | txt | txt | txt | txt | txt |
| \(B_3\) | txt | txt | txt | txt | txt |
| \(C_3\) | txt | txt | txt | txt | txt |
| \(B_4\) | txt | txt | txt | txt | txt |
| \(C_4\) | txt | txt | txt | txt | txt |
| \(D_4\) | txt | txt | txt | txt | txt |
| \(F_4\) | txt | txt | txt | txt | txt |
| \(B_5\) | txt | txt | txt | txt | txt |
| \(C_5\) | txt | txt | txt | txt | txt |
| \(D_5\) | txt | txt | txt | txt | txt |
| \(B_6\) | txt | txt | txt | txt | txt |
| \(C_6\) | txt | txt | txt | txt | txt |
| \(D_6\) | txt | txt | txt | txt | txt |
| \(E_6\) | txt | txt | txt | txt | txt |
| \(B_7\) | txt | txt | txt | txt | txt |
| \(C_7\) | txt | txt | txt | txt | txt |
| \(D_7\) | txt | txt | txt | txt | txt |
| \(E_7\) | txt | txt | txt | txt | txt |
| \(E_8\) | txt | txt | txt | txt | txt |
2. Generalizing Haiman’s Conjecture
pdf Summary
txt Permutations taking the ordering in \(\mathtt{ChevieCharInfo}\) to that in \(\mathtt{ICCTable}\)
Elements \(z \in W\) that duplicate
vectors of Hecke character values \((\chi_x(C'_z))_\chi\):
| \(B_2 = C_2\) | txt |
| \(G_2\) | txt |
| \(B_3 = C_3\) | txt |
| \(B_4 = C_4\) | txt |
| \(D_4\) | txt |
| \(F_4\) | txt |
| \(B_5 = C_5\) | txt |
| \(D_5\) | txt |
Virtual multiplicities \(\alpha_{\psi,
G}^z(x)\) for \(z \in W\) and
\(\psi \in \mathrm{Irr}(W)\), defined
by requiring \[\sum_\psi {\{\chi,
\psi\}}\psi_x(C'_z) = \sum_\psi P_{\psi, \chi}(x)\alpha_{\psi,
G}^z(x)
\quad\text{for all $\chi \in \mathrm{Irr}(W)$},\] where \(\{-, -\}\) is Lusztig’s exotic Fourier
transform
(Ch. 98)
and the \(P_{\psi, \chi}(x)\) are given
by the \(\mathtt{ICCTable}\) for \(G\)
(Ch. 100):
| all \(z\) | rationally smooth \(z\) | |
|---|---|---|
| \(B_2\) | txt | txt |
| \(G_2\) | txt | txt |
| \(B_3\) | txt | txt |
| \(C_3\) | txt | txt |
| \(B_4\) | txt | txt |
| \(C_4\) | txt | txt |
| \(D_4\) | txt | txt |
| \(F_4\) | txt | txt |
| \(B_5\) | txt | txt |
| \(C_5\) | txt | txt |
| \(D_5\) | txt | txt |
| \(B_6\) | txt | |
| \(C_6\) | txt | |
| \(D_6\) | txt | |
| \(E_6\) | txt | |
| \(B_7\) | txt† | |
| \(C_7\) | txt† | |
| \(D_7\) | txt† |
Outputs of test_solved:
| all \(z\) | rationally smooth \(z\) | |
|---|---|---|
| \(B_2\) | txt | txt |
| \(G_2\) | txt | txt |
| \(B_3\) | txt | txt |
| \(C_3\) | txt | txt |
| \(B_4\) | txt | txt |
| \(C_4\) | txt | txt |
| \(D_4\) | txt | txt |
| \(F_4\) | txt | txt |
| \(B_5\) | txt | txt |
| \(C_5\) | txt | txt |
| \(D_5\) | txt | txt |
| \(B_6\) | txt | |
| \(C_6\) | txt | |
| \(D_6\) | txt | |
| \(E_6\) | txt | |
| \(B_7\) | txt† | |
| \(C_7\) | txt† | |
| \(D_7\) | txt† |
† currently restricted to \(z\) of Bruhat length \(\leq 24\)
3. Scripts
To generate raw data from CHEVIE:
| charinfo-icc-families.g | \(\mathtt{CharInfo}\), \(\mathtt{ICCTable}\), and unipotent characters organized \(\mathtt{byFamily}\) |
| all_raw.g | Exotic character values on the Kazhdan–Lusztig basis elements \(C'_z\), for all \(z \in W\) |
| ratlsmooth_raw.g | Exotic character values on the Kazhdan–Lusztig basis elements \(C'_z\), for rationally smooth \(z \in W\) |
To pre-process CHEVIE data:
| join_lines.py* | Joins broken lines in CHEVIE output |
| extract_icc_labels.py* | Extracts \(\mathtt{ICCTable}\) ordering of irreducible characters and local systems |
| find_permutations.py* | Finds permutation taking \(\mathtt{ChevieCharInfo}\) ordering to \(\mathtt{ICCTable}\) ordering |
| csv_families.py* | Converts truncated Lusztig families to CSV format |
| csv_icc.py* | Converts \(\mathtt{ICCTable}\) to CSV format, then evaluates at \(x = 1\) |
| csv_invert.py* | Inverts square matrix of (rational) numbers in CSV format |
To compute the \(\alpha_{\psi, G}^z\)
and test their properties:
| find_duplicates.py* | Finds duplicate vectors of exotic character values |
| permute_vectors.py* | Permutes exotic character values into \(\mathtt{ICCTable}\) ordering |
| csv_multiply.py* | Multiplies vectors of exotic character values by inverse matrices of ungraded Springer multiplicities to obtain \(\alpha_{\psi, G}^z\) |
| test_solved.py* | Tests nonzero coefficients of \(\alpha_{\psi, G}^z\) for unimodal absolute values, mixed signs, or strictly negative signs |
* written with help from Claude Sonnet 4.6