On the semisimplicity and Schur elements of (super)symmetric superalgebras

In this paper, we use Schur elements to derive semisimplicity criteria for (super) symmetric superalgebras. We obtain a closed formula for the Schur elements of cyclotomic Hecke–Clifford superalgebras HᶠK H K f . As applications, we prove that two trace functions ₙ ℷ n and t1,n t 1 , n on the Hecke–Clifford superalgebra, which are defined in different ways, are proportional. We give a semisimplicity criterion for HᶠK H K f when it is (super)symmetric. We also derive semisimplicity criteria for cyclotomic quiver Hecke superalgebras of types A⁽¹⁾ₑ A e ( 1 ) , C⁽¹⁾ₑ C e ( 1 ) , A⁽²⁾₂ₑ A 2 e ( 2 ) and D⁽²⁾ₑ₊₁ D e + 1 ( 2 ) .