Invariant subspaces of the Lawrence–Krammer representation

Abstract

Abstract. The Lawrence-Krammer representation was used in 2000 to show the linearity of the braid group. The problem had remained open for many years. The fact that the Lawrence-Krammer representation of the braid group is reducible for some complex values of its two parameters is now known, as well as the complete description of these values. It is also known that when the representation is reducible, the action on a proper invariant subspace is an Iwahori-Hecke algebra action. In this paper, we prove a theorem of classification for the invariant subspaces of the Lawrence-Krammer space. We classify the invariant subspaces in terms of Specht modules. We fully describe them in terms of dimension and spanning vectors in the Lawrence-Krammer space.

Classification of the Invariant Subspaces of the Cohen–Wales Representation of the Artin Group of TypeDn: Table 1.

“…First, we show that this is true for the vectors W (j) j−2 with 3 ≤ j ≤ n. Second, we show that the result also holds for all the other vectors W (j) k with 3 ≤ j ≤ n and 1 ≤ k ≤ j − 3. To do so, we proceed by descending induction on the integer k. We use the second and fourth equalities of Lemma 3.16 of [8] to derive the relations…”

“…We identify the dimensions of the irreducible H(D n )-modules that may occur in the Cohen-Wales space when n ≥ 11, under some assumption at rank 10. We prove the classification theorem for these n, assuming the classification theorem holds when 4 ≤ n ≤ 10 and we use in particular some results of [8].…”

“…We will use the results of [8] to give a matrix representation for S (0),(n−2,1,1) . In [8] § 2, we give a new representation of the braid group on n strands that is equivalent to the Lawrence-Krammer representation. In that paper, the vectors u 1 , u 2 , u 3 of Theorem 3.5 and the vectors V of the Cohen-Wales space such that the matrices of the left Cohen-Wales actions by the g i 's, 1 ≤ i ≤ n in this basis are the H i 's, 1 ≤ i ≤ n. The left Cohen-Wales action by the g i 's, 1 ≤ i ≤ n on the basis (w ij ) 1≤i<j≤n of the Cohen-Wales space is the Cohen-Wales representation constructed and defined in Theorem 1 of [9] and whose structure is studied in this paper.…”

“…We will use the results of [8] to give a matrix representation for S (0),(n−2,1,1) . In [8] § 2, we give a new representation of the braid group on n strands that is equivalent to the Lawrence-Krammer representation. In that paper, the vectors u 1 , u 2 , u 3 of Theorem 3.5 and the vectors…”

“…It is a result of the paper that the matrices of the left braid group actions by the g i 's, 1 ≤ i ≤ n − 1, in this basis provide a matrix representation for the H F,r 2 (n)-Specht module S (n−2,1,1) . The left braid group actions by the g i 's on the basis of the Lawrence-Krammer space is the braid group representation provided in § 2 of [8]. The braid group actions by the g i 's on the family of vectors (V (k) s ) 5≤k≤n, 1≤s≤n−2 is the object of Lemma 3.16 of [8].…”