Proposition. The positive CAR product has half-staircase weight [fcap-0036]
Proposition. The positive CAR product has half-staircase weight [fcap-0036]
Let \(K\) be a field in which \(2\) is invertible. Begin with \(N=2\) and put \(v_+=d_{01}\). The relations are \[d_{01}^2=d_{10}^2=0,\qquad d_{00}^2=d_{11}^2=1, \qquad d_{01}d_{10}+d_{10}d_{01}=2.\] The only raising operator is \[D(E_{01})=\frac 12(d_{00}d_{01}+d_{01}d_{11}),\] and each summand kills \(v_+\). For the diagonal operators, the same CAR relations give \[D(E_{00})v_+=\frac 32v_+,\qquad D(E_{11})v_+=\frac 12v_+.\] Thus \(v_+\) has weight \((3/2,1/2)\), and \(D(I)v_+=2v_+\). Subtracting the central correction \((N/2)\operatorname {Tr}\) leaves \((1/2,-1/2)\). In characteristic zero this is the usual \(\mathfrak {sl}_2\) value \(\rho \).
For general \(N\), order the indices as \(0<1<\cdots <N-1\) and form the lexicographically ordered product \[v_+=\prod _{0\leq i<j<N}^{\mathrm {lex}}d_{ij}.\] Every positive matrix unit is \(Q_{\mathrm {tr}}\)-null, and distinct positive matrix units are polar-orthogonal. Their Clifford generators therefore anticommute. PBW linear independence shows that the ordered product is nonzero over a field.
Let \(i<j\). In \[D(E_{ij})v_+=\frac 12\sum _kd_{ik}d_{kj}v_+,\] a term with \(k<j\) vanishes because the positive generator \(d_{kj}\) already occurs in \(v_+\). If \(k\geq j\), then \(i<k\); first anticommute \(d_{ik}\) past \(d_{kj}\), using \(i\ne j\), and the repeated positive factor \(d_{ik}\) makes the term zero. Hence every raising operator annihilates \(v_+\).
For a diagonal generator, \[D(E_{ii})v_+=\frac 12\sum _kd_{ik}d_{ki}v_+.\] The contributions for \(k<i\), \(k=i\), and \(k>i\) are respectively \(0\), \(v_+\), and \(2v_+\). Therefore \[D(E_{ii})v_+ =\frac 12\bigl (1+2|\{k:k>i\}|\bigr )v_+.\] For \(i\in \operatorname {Fin}(N)\), the weight and its quadratic-lift part are \[\lambda _i=N-i-\frac 12,\qquad \lambda _i-\frac N2=\frac {N-1}{2}-i,\qquad \sum _i\lambda _i=\frac {N^2}{2}.\] The last equality agrees with the direct central calculation \(D(I)=N^2/2\).
TauCeti defines this ordered product and proves its nonvanishing, raising equations, and diagonal weight. See the exact pinned product and PBW nonvanishing and half-staircase theorems. Nonvanishing needs a field; the highest-vector statement adds invertible \(2\). Identifying the subtracted weight with the rational staircase requires characteristic zero.
[panyushev2001exterior, Section 2, Example 2.5(1), preprint p. 8] identifies, over an algebraically closed field of characteristic zero, the reduced adjoint spin module with highest weight \(\rho \). The subtracted weight above is exactly that semisimple staircase. The added \((N/2)\operatorname {Tr}\) character records the reductive center of this normal-ordered left action. This comparison does not attribute the selected product or the full left-regular CAR module to Panyushev, and it makes no irreducibility or whole-module decomposition claim.