The letter being written, he [modal verb] give you permission.
Can anyone tell me is it okay if I use modal verbs after absolute phrases in the sentence as above?
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I don't think an absolute clause imposes any formal constraint on the verb heading a following matrix clause. An absolute clause is (by definition) not integrated into the syntax of the matrix.
There are of course pragmatic and semantic constraints. Absolutes typically act as what I call ‘scenic’ modifiers on the matrix clause, designating states which obtain at the time of the eventuality in the matrix, so there must be some meaningful relationship between the absolute and the matrix.
But it is quite possible for a matrix clause headed by a modal to be modified by an absolute:
The letter having been submitted in proper form, you may take it for granted that permission will be granted.
His request for a fellowship having been denied, Martin could not afford to continue in school.