While computing some examples in HeckeAlgebraSymmetricGroupT I
discovered some errors.  See the following example

sage: H=HeckeAlgebraSymmetricGroupT(QQ,3)
sage: T1=H.algebra_generators()[0]
sage: T1^-1
T[1, 2, 3]
sage: H.one_element()
T[1, 2, 3]

I would like to take the inverse of the generators in H, and I would
like to know why T1^-1 yields the one_element of the algebra.

My ideal outcome would be that T1^-1 gives q^-1*(T1+1-q).  This leads
to my next question, see the example below:

sage: H=HeckeAlgebraSymmetricGroupT(QQ,3)
sage: q=H.q()
sage: q^-1
1/q
sage: (q^-1)*T1
TypeError: unsupported operand parent(s) for '*': 'Fraction Field of
Univariate Polynomial Ring in q over Rational Field' and 'Hecke
algebra of the symmetric group of order 3 on the T basis over
Univariate Polynomial Ring in q over Rational Field'

Why am I getting this error?

thanks
Jeff
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