Definition. Ordered Berezin coefficients [fdip-008F]

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Let \(V\) be an \(N\)-dimensional vector space over \(\mathbb k=\mathbb R\) or \(\mathbb C\), with an ordered basis \(\xi _1,\ldots ,\xi _N\) of \(V^*\). The Grassmann algebra on these generators is the exterior algebra \(\Lambda (V^*)\). Multiplication is written by juxtaposition, so \[\xi _i\xi _j=-\xi _j\xi _i,\qquad \xi _i^2=0.\] Its basis consists of \(1\) and the increasing words \(\xi _{i_1}\cdots \xi _{i_r}\) with \(i_1<\cdots <i_r\); its dimension is \(2^N\). A word of length \(r\) has degree \(r\). Elements of even degree commute with every homogeneous element; two homogeneous elements of degrees \(r,s\) commute with sign \((-1)^{rs}\).

The Berezin integral with reversed top order is the linear functional \[I_\xi (f)=\text {coefficient of }\xi _N\cdots \xi _1 \text { in }f.\] Thus \(I_\xi (\xi _N\cdots \xi _1)=1\) and it vanishes on all lower degrees. Relative to the increasing top coefficient it has the sign \((-1)^{N(N-1)/2}\). For \(N=0\) it is the identity on \(\mathbb k\). This is the purely odd coefficient definition of [etingof2024mathematical, v2, Definition 9.6, p. 128], with the top order stated explicitly.

For \(n\) complex pairs, take \(2n\) independent generators \(\psi _1,\bar \psi _1,\ldots ,\psi _n,\bar \psi _n\) and use the paired functional \[J_n(f)=\text {coefficient of } \psi _1\bar \psi _1\cdots \psi _n\bar \psi _n \text { in }f.\] The bars label independent generators, not complex conjugates. The symbols \(I_\xi \) and \(J_n\) keep these two top-order conventions distinct. They are coefficient functionals, not positive measures on ordinary field configurations.

For a positive-degree nilpotent element \(q\), define \(e^q=\sum _{r\ge 0}q^r/r!\); only finitely many terms survive. For a quadratic \(q\) in \(N\) generators, terms with \(2r>N\) vanish. No analytic convergence is involved. The ordered monomial basis is supplied formally by Module.Basis.ExteriorAlgebra; this supports the algebraic coordinates, not the Berezin or Gaussian identities.