Reducing Rouquier complexes (Q6581867)

Rouquier complexes, introduced in [\textit{R. Rouquier}, Contemp. Math. 406, 137--167 (2006; Zbl 1162.20301)], are certain complexes defined in the homotopy category of the Hecke category. These complexes define a strong categorical action of the generalised braid group (also known as Artin-Tits group) \(B_W\) associated to the Coxeter system \((W,S)\), which is conjecturally faithful. They are important as many categorical actions of \(B_W\) can be studied as their quotient, and they can also be used to construct a triply-graded link homology via the work of \textit{M. Khovanov} [Int. J. Math. 18, No. 8, 869--885 (2007; Zbl 1124.57003)].\N\NThis paper provides an explicit description of the minimal representatives of these complexes, namely the homotopy equivalent complexes that have no contractible summand (no more Gaussian elimination can be applied). The main tool comes from an algebraic description of (discrete) Morse theory studied in [\textit{E. Sköldberg}, Trans. Am. Math. Soc. 358, No. 1, 115--129 (2006; Zbl 1150.16008)]. The minimal complexes are defined in terms of contraction of monotonous subwords, which can be found in \S 3.6.