The Donald–Flanigan conjecture asserts that the integral group ring
Z
G
\mathbb {Z}G
of a finite group
G
G
can be deformed to an algebra
A
A
over the power series ring
Z
[
[
t
]
]
\mathbb {Z}[[t]]
with underlying module
Z
G
[
[
t
]
]
\mathbb {Z}G[[t]]
such that if
p
p
is any prime dividing
#
G
\#G
then
A
⊗
Z
[
[
t
]
]
F
p
(
(
t
)
)
¯
A\otimes _{\mathbb {Z}[[t]]}\overline {\mathbb {F}_{p}((t))}
is a direct sum of total matric algebras whose blocks are in natural bijection with and of the same dimensions as those of
C
G
.
\mathbb {C}G.
We prove this for
G
=
S
n
G = S_{n}
using the natural representation of its Hecke algebra
H
\mathcal {H}
by quantum Yang-Baxter matrices to show that over
Z
[
q
]
\mathbb {Z}[q]
localized at the multiplicatively closed set generated by
q
q
and all
i
q
2
=
1
+
q
2
+
q
4
+
⋯
+
q
2
(
i
−
1
)
,
i
=
1
,
2
,
…
,
n
i_{q^{2}} = 1+q^{2} + q^{4} + \dots + q^{2(i-1)}, i = 1,2,\dots , n
, the Hecke algebra becomes a direct sum of total matric algebras. The corresponding “canonical" primitive idempotents are distinct from Wenzl’s and in the classical case (
q
=
1
q=1
), from those of Young.