Symplectic Homology Basis Overview

• Symplectic homology basis is a canonical construction in Lie algebra homology, providing explicit wedge-monomials for irreducible Sp(V) representations.
• It utilizes the Chevalley–Eilenberg complex and strict combinatorial rules to identify closed representatives in each homology degree.
• The method underpins representation theory by decomposing modules via single-column Young diagrams and demonstrating Poincaré duality.

A symplectic homology basis is a canonical construction arising in the computation of Lie algebra homology for 2-step nilpotent Lie algebras associated with symplectic groups. Explicitly realized in the context of the symplectic Heisenberg Lie algebra, this basis is governed by the structure of wedge-monomials in a vector space endowed with a nondegenerate skew-symmetric bilinear form. Its elements, selected by stringent combinatorial rules, provide closed representatives for the homology in each degree, yielding irreducible representations of the symplectic group Sp(V). These bases play an essential role in the algebraic and representation-theoretic analysis of homological invariants, as demonstrated in foundational computations by Getzler, Santharoubane, and subsequent developments (Sam, 2013).

1. The Symplectic Heisenberg Lie Algebra

Let $V$ be a complex vector space of dimension %%%%1%%%%, equipped with a nondegenerate symplectic form $\omega : \wedge^2 V \to \mathbb{C}$. A canonical basis $\{e_1, \ldots, e_n, f_1, \ldots, f_n\}$ of $V$ is chosen such that $\omega(e_i, e_j) = 0$, $\omega(f_i, f_j) = 0$, and $\omega(e_i, f_j) = \delta_{ij}$. The Lie algebra $\mathfrak{h} = V \oplus \mathbb{C} z$ is defined by the bracket $[e_i, f_j] = \delta_{ij} z$, with $z$ central; all other brackets vanish. This algebra is a prototypical 2-step nilpotent Lie algebra whose structure constants are dictated by the symplectic form, and its homology encapsulates the representation theory of Sp(V).

2. Chevalley–Eilenberg Complex Construction

The Lie homology of $\mathfrak{h}$ is computed via the Chevalley–Eilenberg complex $C_*(\mathfrak{h}) = \wedge^k \mathfrak{h}$, equipped with the differential

$$d(v_1 \wedge \cdots \wedge v_k) = \sum_{1 \leq a < b \leq k} (-1)^{a+b+1} [v_a, v_b] \wedge v_1 \wedge \cdots \wedge \widehat{v_a} \wedge \cdots \wedge \widehat{v_b} \wedge \cdots \wedge v_k,$$

where $[v_a, v_b]$ refers to the algebra bracket, and hats denote omission. Due to the specific bracket structure, only pairs involving one $e_i$ and one $f_j$ yield nonzero contributions, producing the central element $z$. Generators with $z$ outside such pairs are closed under $d$, solidifying the combinatorial nature of cycle selection.

3. Sp(V)-Module Decomposition and Homology Degrees

The homology $H_k(\mathfrak{h}; \mathbb{C})$ is nonzero precisely for $0 \leq k \leq 2n + 1$. In each degree, it is a single irreducible Sp(V)-module:

• For $0 \leq k \leq n$, $$H_k(\mathfrak{h}) \cong \wedge^k V \cong S_{[1^k]}(V)$$, with $[1^k]$ the single-column Young diagram of length $k$.
• For $n+1 \leq k \leq 2n + 1$, Poincaré duality yields $$H_k(\mathfrak{h}) \cong \wedge^{2n+1 - k} V \cong S_{[1^{2n+1-k}]}(V)$$.

This Sp(V)-module decomposition reflects the combinatorial properties of the wedge monomials constituting the homology basis and ensures representation-theoretic uniqueness in each degree.

4. Explicit Description of Symplectic Homology Bases

A basis for $H_k(\mathfrak{h})$ in degree $k$ takes the following explicit form:

• For $0 \leq k \leq n$: Each basis element is a wedge-monomial $$e_{i_1} \wedge \cdots \wedge e_{i_p} \wedge f_{j_1} \wedge \cdots \wedge f_{j_q}$$, with $p + q = k$ and with the restriction that no index $r$ is shared between $\{i_1, \ldots, i_p\}$ and $\{j_1, \ldots, j_q\}$. This is equivalently described as selecting a subset $S \subseteq \{1, \ldots, n\}$ of size $k$ and for each $i \in S$ choosing either $e_i$ or $f_i$.
• For $n+1 \leq k \leq 2n+1$: Write $m = 2n + 1 - k$. Each basis element is $z \wedge (u_{i_1} \wedge \cdots \wedge u_{i_m})$, where for each $u_{i_a}$, one chooses either $e_{i_a}$ or $f_{i_a}$, again with no repeated index.

All such monomials are closed, and the selection rule forbids the simultaneous appearance of $e_i$ and $f_i$ for any fixed $i$ within a monomial.

5. Combinatorial Selection Rule

The core combinatorial principle stipulates that a wedge-monomial in the alphabet $\{e_i, f_i\}$, of total degree $k$, survives in homology if and only if it does not contain both members of any symplectic pair $\{e_i, f_i\}$. In the language of Young diagrams, such a monomial is associated with a single-column diagram whose length equals the number of distinct indices used, matching either $k$ or $2n+1-k$, thereby identifying the correct Sp(V)-irreducible.

6. Low-Dimensional Illustrations

Illustrative examples for small $n$ make the selection mechanism concrete:

Case $n=1$:

$k$	$H_k(\mathfrak{h})$	Basis Elements
0	$\mathbb{C}$	$1$
1	$V$	$e$, $f$
2	$V$	$z \wedge e$, $z \wedge f$
3	$\mathbb{C}$	$z \wedge e \wedge f$

Case $n=2$:

$k$	$H_k(\mathfrak{h})$	Basis Elements
0	$\mathbb{C}$	$1$
1	Span$\{e_1,e_2,f_1,f_2\}$	$e_1$, $e_2$, $f_1$, $f_2$
2	Span$\{e_1 \wedge e_2,\ldots\}$	$e_1 \wedge e_2$, $e_1 \wedge f_2$, $e_2 \wedge f_1$, $f_1 \wedge f_2$
3	Span$\{z \wedge e_1,\ldots\}$	$z \wedge e_1$, $z \wedge e_2$, $z \wedge f_1$, $z \wedge f_2$
4	Span$\{z \wedge e_1 \wedge e_2,\ldots\}$	$z \wedge e_1 \wedge e_2$, $z \wedge e_1 \wedge f_2$, $z \wedge e_2 \wedge f_1$, $z \wedge f_1 \wedge f_2$
5	$\mathbb{C}$	$z \wedge e_1 \wedge e_2 \wedge f_1 \wedge f_2$

These explicit bases are illustrative of the selection rule and duality between homology degrees.

7. Connections to Representation Theory and Related Computations

The symplectic homology basis construction, via combinatorial methods and Sp(V)-module decomposition, completes foundational calculations initiated by Getzler and Santharoubane, employing both direct analysis and representation-theoretic frameworks. The symmetry between degrees via Poincaré duality and the manifestation of highest weights indexed by single-column Young diagrams highlight deep links to Weyl group combinatorics and classical representation theory. The general methodology extends to homology computations for other families of nilpotent Lie (super)algebras associated with orthogonal and general linear groups via analogous combinatorial and representation-theoretic techniques (Sam, 2013).