The nil-Hecke divided-power projective in rank two

(a) Using [F1], $e_{i,2}^2=x_1\tau x_1\tau =x_1(x_2\tau +e)\tau =x_1{x}_2{\tau }^2+x_1\tau =x_1\tau =e_{i,2}$ because ${\tau }^2=0$, and $e_{i,2}\mathrm{\ne }0$ because $x_1\tau$ is a basis monomial; its degree is $2d_i+(-2d_i)=0$. Likewise $\psi (e_{i,2}{)}^2=\tau x_1\tau x_1=(x_2\tau +e)\tau x_1=x_2{\tau }^2{x}_1+\tau x_1=\tau x_1$, so $\psi (e_{i,2})$ is an idempotent of degree $0$. The left ideal $R(2{\alpha }_i)\psi (e_{i,2})$ is the polynomial representation: the map $a\psi (e_{i,2})\mapsto a\cdot 1$ is surjective onto $k[x_1,x_2]$ because $\psi (e_{i,2})1=1$, and its inverse sends $f$ to $f\tau x_1$; the two maps are inverse because $\psi (e_{i,2})$ is the rank-one projection exhibiting $N{H}_2={\mathrm{End}}_{k[x_1,x_2{]}^{S_2}}(k[x_1,x_2])$, so the module is indecomposable projective and its homogeneous endomorphism algebra is the symmetric polynomial ring $k[x_1,x_2{]}^{S_2}$, whose degree-zero part is $k$. As a projective with simple head it is the projective cover of the unique graded simple column $k[x_1,x_2]\mathrm{/}(x_1+x_2,x_1{x}_2)$, of dimension $2$, up to shift. The positive symmetric-center ideal gives its simple quotient; its kernel is graded superfluous by the least-degree argument, or by the matrix equivalence with the connected positively graded center; the displayed shift $-d_i$ is the one fixed in [F2] and does not affect these statements except by transport.

(b) The endomorphism ring computation of step 1.1 shows $R(2{\alpha }_i)\cong {\mathrm{End}}_{k[x_1,x_2{]}^{S_2}}(k[x_1,x_2])$ as a graded module over itself, and this is the direct sum of $2=[2{]}_i!{\mathrm{\mid }}_{q_i=1}$ shifted copies of the polynomial column $P_{i,2}$: the polynomial ring is free of rank $2$ over its symmetric subring with basis $\{1,x_1\}$ (or the Artin monomials $\{1,x_1\}$ for $m=2$), so the duals of the two basis monomials have degrees $0,-2d_i$, so the regular module is $V\oplus V\{-2d_i\}$ for the polynomial column $V$. Since $P_{i,2}=V\{-d_i\}$, this is $P_{i,2}\{d_i\}\oplus P_{i,2}\{-d_i\}$. By [F4] the decomposition into indecomposables is unique up to order and shift, so $P_{i,2}$ is the unique indecomposable projective summand up to shift. The graded multiplicity in the normalization of [F2] is $[2{]}_{q_i}!=[1{]}_{q_i}[2{]}_{q_i}=1\cdot (q_i+q_i^{-1})$, the value recorded in the statement, and at $q_i=1$ it is the ungraded multiplicity $2$; the two polynomial columns carry the permutation shifts of the two Artin monomials and are absorbed by this normalization.

(c) The class identification $[P_{i,2}]=\gamma ({\theta }_i^{(2)})$ is [F2] with $n=2$. The self-pairing is the rank-$n$ value of [F3] at $n=2$: $([P_{i,2}],[P_{i,2}])={\prod }_{a=1}^2(1-q_i^{2a}{)}^{-1}=(1-q_i^2{)}^{-1}(1-q_i^4{)}^{-1}$, which is the Lusztig-normalized value $B_L({\theta }_i^{(2)},{\theta }_i^{(2)})$ by the form comparison.