Nil-Hecke idempotents and divided-power KLR projectives

Let $Q$ be a KLR datum as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum. For $i\in I$ and $n\ge 1$ the subalgebra $R(n{\alpha }_i)$ is generated by the polynomial subalgebra $k[x_1,\dots ,x_n]$ and the crossings ${\tau }_1,\dots ,{\tau }_{n-1}$ with ${\tau }_a^2=0$ (because $Q_{ii}=0$), the braid and far commutation relations, and the nil-Hecke dot moves, i.e. it is the nil Hecke algebra of rank $n$; its PBW basis is $\{x_1^{a_1}\cdots x_n^{a_n}{\tau }_w{\}}_{a\in {\mathbb{Z}}_{\ge 0}^n,w\in S_n}$ (The PBW (monomial) basis theorem for KLR algebras). Fix a reduced expression $w_0=s_{i_1}\cdots s_{i_{\left(\genfrac{}{}{0px}{}{n}{2}\right)}}$ of the longest element and put ${\mathrm{\partial }}_{w_0}:={\tau }_{i_1}\cdots {\tau }_{i_{\left(\genfrac{}{}{0px}{}{n}{2}\right)}}$ (independent of the reduced expression). The divided-power idempotent is $e_{i,n}:=x_1^{n-1}{x}_2^{n-2}\cdots x_{n-1}{\mathrm{\partial }}_{w_0}\in R(n{\alpha }_i),$ and its horizontal reflection is $\psi (e_{i,n}):={\mathrm{\partial }}_{w_0}{x}_1^{n-1}\cdots x_{n-1}$. Both are nonzero idempotents of degree $0$, and $e_{i,n}\mathrm{\ne }0$ because the PBW basis separates the monomials involved. The divided-power projective is $P_{i,n}:=R(n{\alpha }_i)\psi (e_{i,n})\{-d_i\left(\genfrac{}{}{0px}{}{n}{2}\right)\},$ where the shift is the one fixed by the source normalization: as a graded left module $R(n{\alpha }_i)\psi (e_{i,n})$ is the polynomial representation, it is indecomposable projective, and with this shift $R(n{\alpha }_i)$ is a finite direct sum of $[n{]}_i!$ shifted copies of $P_{i,n}$, where $[n{]}_i!={\prod }_{a=1}^n\frac{q_i^a-q_i^{-a}}{q_i-q_i^{-1}}$ is the $q_i$-factorial of Quantum integers, factorials, Gaussian binomials and divided powers at $q_i$ (at $q_i=1$ it is $n!$); hence $P_{i,n}$ is the unique indecomposable projective summand up to shift and ${\mathrm{E}\mathrm{n}\mathrm{d}}_{R(n{\alpha }_i)}(P_{i,n}{)}_0=k$. For a divided-power sequence $\theta =i_1^{(n_1)}\cdots i_r^{(n_r)}$ put $\beta ={\sum }_a{n}_a{\alpha }_{i_a}$ and $P_{\theta }:=\mathrm{I}\mathrm{n}\mathrm{d}(P_{i_1,n_1}⊠\cdots ⊠P_{i_r,n_r})$ along the ordered concatenation of the blocks of Induction and restriction bimodules for KLR algebras; then $[P_{\theta }]$ represents the ordered product of these divided powers; one may reorder only by the proved commuting/Serre relations, adjacent colours need not commute, and for a single colour $\theta =i^{(n)}$ one has the indecomposability above.

Projective duality. For a finite graded projective left $R(\beta )$-module $P$ put $P^{\mathrm{\#}}:={\mathrm{HOM}}_{R(\beta )}(P,R(\beta ){)}^{\psi }$, the $\psi$-twisted graded Hom into the rank-one free module, where $\psi$ is the horizontal-reflection anti-involution. Evaluation $P^{\mathrm{\#}}\to$ (dual of $P$) identifies $(P^{\mathrm{\#}}{)}^{\mathrm{\#}}\cong P$ for finite shifted free modules, and hence for every finite graded projective; the duality reverses internal shifts and fixes each normalized $P_{i,n}$ after the displayed shift.

Nil-Hecke matrix calculation. Put $V=k[x_1,\dots ,x_n]$, $C=k[e_1,\dots ,e_n]=V^{S_n}$ and $N=\left(\genfrac{}{}{0px}{}{n}{2}\right)$. The elementary symmetric functions are algebraically independent and generate the symmetric ring: subtracting the leading symmetric monomial orbit successively expresses every invariant as a polynomial in them, and their distinct leading monomials give independence. Adjoin $x_1$ as a root of the monic polynomial $T^n-e_1{T}^{n-1}+\cdots +(-1{)}^n{e}_n$, then $x_2$ as a root of its quotient by $T-x_1$, and continue. Eliminating coefficients recovers the polynomial ring in the roots; each monic extension is free on powers below its degree. Hence $V$ is free over $C$ on $x^a$ with $0\le a_r\le n-r$.

Write $\delta =(n-1,n-2,\dots ,0)$. Expanding the divided differences gives ${\mathrm{\partial }}_{w_0}(f)={\sum }_{w\in S_n}\mathrm{sgn}(w)w(f)\mathrm{/}{\prod }_{a<b}(x_a-x_b)$: the braid relations make this independent of the reduced word, and the numerator of $x^{\delta }$ is the Vandermonde determinant. Thus ${\mathrm{\partial }}_{w_0}(x^{\delta })=1$, its values are symmetric, and it kills every other Artin basis monomial by degree. It is therefore the coefficient-of-$x^{\delta }$ trace of the monic tower. This trace has a perfect $C$-bilinear pairing $⟨f,g⟩={\mathrm{\partial }}_{w_0}(fg)$: for a monic extension of degree $m$, the coefficient-of-$T^{m-1}$ trace has matrix zero below the antidiagonal and ones on it in the power basis, hence determinant $\pm 1$; composing these perfect trace pairings through the tower preserves perfection. Choose the finite trace-dual Artin basis $(b_a^{\ast })$ to $(b_a)$. The operators $f\mapsto b_a{\mathrm{\partial }}_{w_0}(b_b^{\ast }f)$ are matrix units, all lying in $N{H}_n$; faithfulness of Faithful polynomial representation of a KLR algebra therefore identifies $N{H}_n={\mathrm{End}}_C(V)$ in every characteristic.

Idempotents, column and duality. In this faithful action, $E={\mathrm{\partial }}_{w_0}{x}^{\delta }$ sends $f$ to ${\mathrm{\partial }}_{w_0}(x^{\delta }f)\in C$ and fixes $C$, so $E^2=E$. Reflection $\psi$ fixes dots and crossings and reverses products, giving $e_{i,n}=\psi (E)$ also idempotent and nonzero. Both have degree zero. The map $N{H}_{n}E\to V$, $aE\mapsto a(1)$, is an isomorphism with inverse $f\mapsto m_{f}E$: $E$ is the rank-one projection onto $C\cdot 1$, so $aE=m_{a(1)}E$. Its endomorphism ring is $C$ and its degree-zero part is $k$, proving graded indecomposability. The trace pairing has degree $-2d_{i}N$ and satisfies $⟨af,g⟩=⟨f,\psi (a)g⟩$: this is immediate for dots and follows for ${\mathrm{\partial }}_r$ by expanding $(1-s_r)\mathrm{/}(x_r-x_{r+1})$ in the trace formula. Consequently $V^{\mathrm{\#}}\cong V\{-2d_{i}N\}$, and $P_{i,n}=V\{-d_{i}N\}$ is self-dual. The Artin basis degrees yield $N{H}_n\cong {⨁}_{a}V\{-2d_i\mathrm{\mid }a\mathrm{\mid }\}$, since ${\mathrm{End}}_C(V)=V{\otimes }_C{V}^{\ast }$; centering gives multiplicity $q_i^N{\prod }_{m=1}^n(1+q_i^{-2}+\cdots +q_i^{-2(m-1)})=[n{]}_i!$. This also proves uniqueness of the indecomposable projective shift orbit: a finite graded projective over the connected positively graded ring $C$ is shifted-free, by lifting a basis modulo $C^{+}$ and using its projective splitting and the least-degree argument on kernel and cokernel, and the matrix equivalence transports that assertion to $N{H}_n$.

Duality and induction. For a finite graded projective $B=R(\beta )\otimes R(\gamma )$-module $M$, finite-projective evaluation gives ${\mathrm{HOM}}_R(Re{\otimes }_{B}M,R)\cong {\mathrm{HOM}}_B(M,eR)\cong {\mathrm{HOM}}_B(M,B){\otimes }_{B}eR$. Twisting by reflection yields $(\mathrm{Ind}M{)}^{\mathrm{\#}}\cong \mathrm{Ind}(M^{\mathrm{\#}})$ with the same block order and grading. The identities are checked on finite shifted-free $M$ and then its summands. Thus ordered inductions of the normalized one-color columns are self-dual, as required for bar compatibility.

Two colours and order. For several colours $P_{\theta }$ is defined by induction along the ordered blocks, so the class is the ordered induction product; blocks of orthogonal colors commute by the invertible degree-zero block shuffle, since their $Q_{ij}$ are nonzero constants, while adjacent colours are related only by the Serre relations proved later on this page, so no silent symmetrization of the word $\theta$ is made. The construction selects no representatives beyond the fixed reduced-expression system of The PBW (monomial) basis theorem for KLR algebras and uses no choice principle.