\(q\)-identities and affinized projective varieties. I: Quadratic monomial ideals (Q1976011)

An algorithm to associate a \(q\)-identity (cf.\ \textit{infra}) to a projective variety \(V\) is given when \(V\) is defined by a quadratic monomial. The identity arises by calculating the Hilbert series of the (homogeneous) coordinate ring of so-called affinized projective varieties. Let \(V\subset{\mathbb P}^{n-1}\) be a (complex) projective variety defined by the ideal \({\mathcal I}(V)\) generated by a set of homogeneous elements \(f_i\), \(i=1,\ldots,t\). The affinized projective variety \(\widehat{V}\subset\widehat{{\mathbb P}^{n-1}}\) is the infinite dimensional projective variety defined by the ideal \({\mathcal I}(\widehat{V})\) generated by the relations \(f_i[m]\), \(i=1,\ldots,t\), \(m\in{\mathbb Z}_{\geq 0}\), in \(\widehat{{\mathcal S}}={\mathbb C}[x_1,\ldots,x_n]\otimes{\mathbb C}[[t]]={\mathbb C}[x_1[m],\ldots,x_n[m]]_{m\in{\mathbb Z}_{\geq 0}}\), where \(x_i[m]=x_i\otimes t^m\) and where \(f_i[m]\) is obtained from \(f_i\) by replacing the monomials \(x_{i_1}\ldots x_{i_r}\) by \[ (x_{i_1}\ldots x_{i_r})[m]=\sum_{\substack{ n_1,\ldots,n_r\\ n_1+\cdots +n_r=m}} x_{i_1}[n_1]\ldots x_{i_r}[n_r]. \] The coordinate ring \({\mathcal S}(\widehat{V})\) is graded by the multidegree given by \(\deg(x_i[m])=(\deg(x_i):m)\). Let \({\mathcal S}(\widehat{V})_{(M:N)}\) be the vector space of polynomials \(f\) of multidegree \((M:N)\) in \({\mathcal S}(\widehat{V})\). The Hilbert function is defined as \(h_{\widehat{V}}(M:N)=\dim{\mathcal S}(\widehat{V})_{(M:N)}\). The Hilbert series of \(\widehat{V}\) is defined by \(h_{\widehat{V}}(y:q)=\sum_{M,N}h_{\widehat{V}}(M:N)y^Mq^N\). One also has the partial Hilbert series \({h_{\widehat{V}}(M:q)=\sum_{N}h_{\widehat{V}}(M:N)q^N}\). As a prime example one may consider the ideal \(I=(x_1x_2)\subset{\mathbb C}[x_1,x_2]\) and the corresponding projective variety \(V\). The associated affinized projective variety is given by the ideal \(\widehat{I}\subset{\mathbb C}[x_1[m],x_2[m]]= \widehat{{\mathcal S}}\) generated by all \(f[m]\), \(m\in{\mathbb Z}_{\geq 0}\), where \(f[m]=(x_1x_2)[m]= \sum_{r+s=m}x_1[r]x_2[s]\). One almost immediately finds that \[ h_{\widehat{V}}(M_1,M_2;q)={q^{M_1M_2}\over{(q)_{M_1}(q)_{M_2}}}, \] where \((q)_N=\prod_{k=1}^N(1-q^k)\). In this case one may construct a resolution of \({\mathcal S}(\hat{V})\) (\(\hat{V}\) is a complete intersection) and apply the Euler-Poincaré principle to obtain \[ h_{\hat{V}}(M_1,M_2:q)=\sum_{m\geq 0}(-1)^m{q^{{1\over 2}m(m-1)} \over{(q)_m(q)_{M_1-m}(q)_{M_2-m}}}. \] Comparing both expressions for \(h_{\hat{V}}(M_1,M_2:q)\) gives a basic \(q\)-identity. This procedure may be extended to more general quadratic monomial ideals. The cases of \(I=(x_1x_2,x_2x_3)\subset{\mathbb C}[x_1,x_2,x_3]\), and \(I=(x_1x_2,x_2x_3,x_1x_3)\subset{\mathbb C}[x_1,x_2,x_3]\) are worked out in detail. Also, generalizations to \(I=(x_1x_2,\ldots,x_{n-1}x_n)\subset{\mathbb C}[x_1,\dots,x_n]\) and \(I=(x_1x_2,\ldots,x_{n-1}x_n,x_1x_n)\subset{\mathbb C}[x_1,\ldots,x_n]\) are studied. The resulting identities can be applied to the description of systems of quasi-particles containing null-states and they can then be interpreted as alternating sums of quasi-particle Fock space characters.