THE CHAMBER COMPLEX FOR THE
LITTLEWOOD-RICHARDSON COEFFICIENTS OF GL4.
EMMANUEL BRIAND, MERCEDES ROSAS, AND STEFAN TRANDAFIR
The Li lewood-Richa dson coe icien s cν
λ,µ a e a amily o cons an s o cen al
impo ance in ep esen a ion heo y. They a e indexed by iples o in ege pa -
i ions. I is known ha when he size o he pa i ions is limi ed o some ixed
in ege N(coe icien s associa ed o GLN), hese coe icien s a e gi en by piecewise
polynomial o mulas in λ1,...λN, µ1, . . . , µN, ν1, . . . , νN−1. Mo e p ecisely, he do-
mains o polynomiali y a e he big cells (”chambe s”) o a subdi ision o a cone
in R3N−1( he ”chambe complex”). In 2005, ´
E. Rassa has calcula ed explici ly
he chambe complex o N= 3, inding 18 chambe s subdi iding a cone in R8,
and he co esponding polynomial o mulas (in his case, linea ). Using Rassa ’s
desc ip ion, E. B iand and M. Rosas ecen ly lis ed exhaus i ely all symme ies o
he chambe complex o N= 3, inding a g oup o 288 symme ies (much mo e
ha he 24 symme ies known o exis o gene al N). I is na u al o look a
wha happens in he nex case, N= 4. Wi h his aim, we ha e compu ed (using
Sagema h and Singula ) he chambe complex o N= 4. This objec is la ge,
since i consis s in he subdi ision o a cone in R11 in 67769 chambe s. This compu-
a ion is based on known combina o ial desc ip ions o he Li lewood-Richa dson
coe icien s. Fo ins ance, he Li lewood-Richa dson coe icien c(26,21,14,11)
(16,13,5,2),(17,12,6,1)
coun s he iangles
36
34 53
29 x365
16 x1x271
0 26 47 61 72
illed wi h in ege s subjec o he inequali ies a+b≥c+din all hombi:
d
b c a b c b
d a c a d
o he iangle. (The bo om bo de en ies 0,26,47,61,72 a e he cumula ed sums
o (26,21,14,11), ha a e: 0,26,26+21,26+21+14,26+21+14+11 and he op
bo de en ies 0,16,29,34,36,5365,71,72 a e he cumula ed sums o (16,13,5,2)
and (17,12,6,1), ha a e 0,16,16+13,16+13+5,...,16+13+5+2+17+12+6+1).
E. B iand, Depa amen o Ma em´
a ica Aplicada I, Uni e sidad de Se illa
M. Rosas, Depa amen o de ´
Algeb a, Uni e sidad de Se illa
S. T anda i , Simon F ase Uni e si y.
E. B iand and M.Rosas a e pa ially suppo ed by MTM2016-75024-P and FEDER, and Jun a
de Andalucia unde g an FQM333.