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We show that Lawrence's representation and linear representations fromquantum sl_2 called generic highest weight vectors detect the dual Garsidelength of braids in a simple and natural way. That is, by expressing arepresentation as a matrix over a Laurent polynomial ring using certain naturalbasis, the span of the variable is equal to the constant multiples of the dualGarside length.

Reading the dual Garside length of braids from homological and quantum representations