Witten zeta function - Alchetron, The Free Social Encyclopedia

In mathematics, the Witten zeta function, introduced by Witten (1991), is a function associated to a root system that encodes the degrees of the irreducible representations of the corresponding Lie group. It is a special case of the Shintani zeta function.

Definition

Witten's original definition of the zeta function of a semisimple Lie group was

∑ R 1 dim ⁡ ( R ) s

where the sum is over equivalence classes of irreducible representations R.

If Δ of rank r is a root system with n positive roots in Δ⁺ and with simple roots λ_i, the Witten zeta function of several variables is given by

ζ W ( s 1 , … , s n ) = ∑ m 1 , … , m r > 0 ∏ α ∈ Δ + 1 ( α ∨ , m 1 λ 1 + ⋯ + m r λ r ) s α ,

The original zeta function studied by Witten differed from this slightly, in that all the numbers s_α are equal, and the function is multiplied by a constant.

References

Witten zeta function Wikipedia

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