2011/10/01 by Marcus Bishop, J. Matthew Douglass, Bishop, Marcus +4 · 1 citation
Computer Science · Mathematics · #20C15 #20C40 #20F55 #52C35 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1110.0120
openalex publication_date 2011/10/01 · openalex created_date 2022/09/20 · openalex updated_date 2026/07/28
In recent papers we have refined a conjecture of Lehrer and Solomon\nexpressing the characters of a finite Coxeter group W afforded by the\nhomogeneous components of its Orlik-Solomon algebra as sums of characters\ninduced from linear characters of centralizers of elements of W. Our refined\nconjecture also relates the Orlik-Solomon characters above to the terms of a\ndecomposition of the regular character of W related to the descent algebra of\nW. A consequence of our conjecture is that both the regular character of W\nand the character of the Orlik-Solomon algebra have parallel, graded\ndecompositions as sums of characters induced from linear characters of\ncentralizers of elements of W, one for each conjugacy class of elements of\nW. The refined conjecture has been proved for symmetric and dihedral groups.\nIn this paper we develop algorithmic tools to prove the conjecture\ncomputationally for a given finite Coxeter group. We use these tools to verify\nthe conjecture for all finite Coxeter groups of rank three and four, thus\nproviding previously unknown decompositions of the regular characters and the\nOrlik-Solomon characters of these groups.\n