Integer-color degenerate Hecke–KLR local inverse tables

The coefficient completions. A symmetric polynomial in the commuting $x$ is central: the mixed relation gives $s_{r}f=({\sigma }_{r}f)s_r-(f-{\sigma }_{r}f)\mathrm{/}(x_r-x_{r+1})$, first on monomials by induction, then on polynomials. A symmetric $f$ has zero correction. The fiber of the elementary symmetric polynomials over their values on $\beta$ consists exactly of the ordered permutations of $\beta$: the coordinates are roots with multiplicities of ${\prod }_{a\in \beta }(t-a)$. Its radical is the intersection of the finitely many corresponding maximal ideals, by the polynomial factorization and the fact that every coordinate satisfies this monic polynomial. The elementary-symmetric ideal and this radical have cofinal powers, since the fiber is finite dimensional and its radical nilpotent. Chinese remainders and completion therefore give ${\mathcal{B}}_{\beta }$ with local coordinates $x_r-{\nu }_r$. On the KLR side symmetric dot polynomials are central by [F1], and the same argument at the all-zero fiber shows that its coefficient completion is ${\mathcal{B}}_{\beta }$.

Invertible tables. If $a\mathrm{\ne }b$ the integer $a-b$ is a nonzero unit in $k$, so $z^{-1}$ exists as a formal geometric series. In the backward case $a=b+1$, $q=z^{-1}$ has constant $1$. In the other unequal cases $q=1-z^{-1}$ has constant $1-(a-b{)}^{-1}$, which could vanish only when $a-b=1$, already excluded. Equal colors give constant $1$. For unequal colors $p_{ba}(v,u)=-p_{ab}(u,v)$. Direct multiplication gives $q_{ab}(u,v)q_{ba}(v,u)=1-z^{-2}$ on nonadjacent colors; on a forward arrow it gives $-(z-1)\mathrm{/}{z}^2=(1-z^{-2})\mathrm{/}(v-u)$ since $z=-1+u-v$; on a backward arrow it gives $(z+1)\mathrm{/}{z}^2=(1-z^{-2})\mathrm{/}(u-v)$ since $z=1+u-v$. Thus $Q_{ab}{q}_{ab}\sigma (q_{ba})=1-p_{ab}^2$ in all unequal cases, with no division by a nonunit left unresolved.

A concrete KLR action. On ${\mathcal{B}}_{\beta }$, dots multiply and idempotents project. Define $E_r=({\sigma }_r-1)\mathrm{/}(y_r-y_{r+1})$ on equal adjacent colors. The numerator is divisible by the denominator for polynomials and formal series, so $E_r$ preserves the coefficient algebra. For unequal source colors $(a,b)$ define ${\psi }_r(fe(\nu ))=c_{ab}(y_r,y_{r+1}){\sigma }_r(f)e(s_r\nu )$, where $c_{ab}(u,v)=v-u$ if $a=b+1$ and $c_{ab}(u,v)=1$ otherwise. These explicit operators prove the local relations in the following five cases.

Quadratic and dot moves. Write $\delta =u-v$. On equal colors $E^2=0$, since $E(f)$ is invariant under $\sigma$, and $Eu-vE=-1$, $Ev-uE=1$ follow by expanding the numerator. On unequal colors the dot moves are permutation identities and ${\psi }^2=c_{ba}(u,v)c_{ab}(v,u)=Q_{ab}(u,v)$. Indeed these products are $v-u$ on a forward arrow, $u-v$ on a backward arrow, and $1$ off edges. Disjoint crossings commute because their variables and their two-color factors are disjoint. Idempotent moves and commutative dot relations hold by construction.

Three distinct colors. Put $u=y_1,v=y_2,w=y_3$, $s={\sigma }_1,t={\sigma }_2$, $\rho =sts=tst$, $L={\psi }_1{\psi }_2{\psi }_1$, $M={\psi }_2{\psi }_1{\psi }_2$. On a source $(a,b,c)$ with all colors distinct, direct consecutive application of step 3.1 gives both $L(f)$ and $M(f)$ equal to $c_{bc}(u,v)c_{ac}(u,w)c_{ab}(v,w)\rho (f)$ in the $(c,b,a)$ summand. The three pair factors commute. Hence $L=M$ for every edge pattern among three distinct integer colors.

Exactly the first two equal. On $(a,a,b)$, $a\mathrm{\ne }b$, put $h=c_{ab}(u,v)c_{ab}(u,w)$. Consecutive application gives $L(f)=hst{E}_1(f)$ and $M(f)=E_2(hstf)$. The polynomial $h$ is symmetric in $v,w$, including the backward case $h=(v-u)(w-u)$, so the product rule $E_2(hg)=E_2(h)g+t(h)E_2(g)$ gives $M(f)=h{E}_{2}stf$. Finally $E_{2}st=st{E}_1$: both sides equal $(\rho -st)\mathrm{/}(v-w)$, with the denominator transformed when the permutation lies to its left. Thus $L=M$.

Exactly the last two equal. On $(a,b,b)$, $a\mathrm{\ne }b$, put $h=c_{ab}(u,w)c_{ab}(v,w)$. The two words give $L(f)=E_1(htsf)$ and $M(f)=hts{E}_2(f)$. Here $h$ is symmetric in $u,v$, including $h=(w-u)(w-v)$ when $a=b+1$. The product rule gives $L(f)=h{E}_{1}tsf$. The identity $E_{1}ts=ts{E}_2=(\rho -ts)\mathrm{/}(u-v)$ now yields $L=M$. This explicitly checks the second repeated-neighbor pattern.

Equal outer colors. On $(a,b,a)$, $a\mathrm{\ne }b$, full expansion of the single middle divided difference gives $L(f)=c_{ba}(u,v)[c_{ab}(v,w)\rho (f)-c_{ab}(v,u)f]\mathrm{/}(u-w)$ and $M(f)=c_{ab}(v,w)[c_{ba}(u,v)\rho (f)-c_{ba}(w,v)f]\mathrm{/}(u-w)$. The $\rho (f)$ terms cancel, and step 4.1 gives $(L-M)(f)=[Q_{ab}(w,v)-Q_{ab}(u,v)]f\mathrm{/}(u-w)$. Thus the correction is $+1$ for $a\to b$, $-1$ for $a\leftarrow b$, and zero off edges. The displayed expressions have removable denominators because their original numerators come from divided differences; the final quotient is a polynomial.

All three equal. For $\mathrm{\Delta }=(u-v)(u-w)(v-w)$, expand the three factors $E=(\sigma -1)\mathrm{/}(\text{dot difference})$ in either order. Moving permutations to the right gives the complete identity $E_1{E}_2{E}_1=E_2{E}_1{E}_2=(\rho -st-ts+s+t-1)\mathrm{/}\mathrm{\Delta }$. For example the $\rho$ coefficient is $1\mathrm{/}\mathrm{\Delta }$, the $st$ and $ts$ coefficients are $-1\mathrm{/}\mathrm{\Delta }$, the $s,t$ coefficients are $1\mathrm{/}\mathrm{\Delta }$, and the identity coefficient is $-1\mathrm{/}\mathrm{\Delta }$. Both sides preserve formal series by their original divided-difference formulas, so this rational equality proves their equality without inverting dots in the algebra. This supplies the all-equal case omitted in the cited source.

Faithfulness for the KLR action. Moving dots to the right using step 4.1, and replacing repeated crossings and braid differences using steps 5.1–9.1, spans the presented KLR algebra by ${\psi }_{w}f(y)e(\nu )$ for fixed reduced words $w$. Termination uses crossing length: every quadratic reduction or braid correction has smaller length; reduced words for a permutation are connected by the elementary symmetric-group braid moves. In the rational-function action each ${\psi }_w$ has leading term a nonzero coefficient times $w$, and all other terms have smaller permutation length. Distinct permutations of variables are linearly independent over the rational-function field: a shortest putative relation, evaluated on $fg$ minus one selected automorphism's value on $g$ times its evaluation on $f$, gives a shorter nonzero relation. Hence these spanning monomials are independent. The same triangular argument over fractions of formal series proves faithfulness of the completed action, whose elements are the finite sum over $w,\nu$ with formal coefficient $f$.

Why formal extension is legitimate. There are finitely many $w,\nu$, and each basis vector ${\psi }_{w}e(\nu )$ has a finite crossing degree. Symmetric positive-degree dots form a central ideal with powers cofinal with dot-degree powers in each coefficient algebra by step 1.1. The PBW module just proved is consequently finite free over its right coefficient algebra and completes coefficientwise. Multiplication extends continuously: moving a formal coefficient through a crossing uses a variable permutation or one divided difference, decreasing its order by at most one, so any finite product has a uniform finite order loss. The displayed completed PBW expansion is therefore the actual central completion, rather than an additional relation imposed on $R_{\beta }$.

The common degenerate Hecke operator. In the fraction algebra set $Z_r=x_r-x_{r+1}$ and $S_r=(1+Z_r^{-1}){\sigma }_r-Z_r^{-1}$. On an unequal source component the diagonal term is $-z^{-1}f$, while its crossing coefficient on the target component is $1+(b-a+u-v{)}^{-1}$. On equal colors this operator is ${\sigma }_r-(1-{\sigma }_r)\mathrm{/}(u-v)$ and preserves formal series. Thus $S_r$ acts on ${\mathcal{B}}_{\beta }$ in every case. All identities in the next two steps are computed in the fraction algebra and then hold for these regular operators. This use of auxiliary fractions does not assert that a dot difference is invertible in the completion or its quotients.

Hecke quadratic and mixed identities. Let $A=Z_r^{-1}$; then ${\sigma }_r(A)=-A$. Expanding $S_r^2$ gives identity coefficient $(1+A)(1-A)+A^2=1$ and crossing coefficient $(1+A)A-A(1+A)=0$. Thus $S_r^2=1$, including equal colors. In $S_r{x}_{r+1}-x_r{S}_r$ the crossing terms cancel and the diagonal term is $A(x_r-x_{r+1})=1$. Disjoint $S_r$ commute, and dots outside the pair commute with $S_r$. These are precisely the quadratic, mixed and far-commuting relations of [F2].

Complete Hecke braid coefficients. In spectral variables put $Z=x_1-x_2$, $U=x_2-x_3$, $V=Z+U$, $A=Z^{-1}$, $B=U^{-1}$, $C=V^{-1}$. Both products $S_1{S}_2{S}_1$ and $S_2{S}_1{S}_2$ have coefficients, in permutation order $(1,s,t,st,ts,\rho )$, equal to $(-C-ABC,(1+A)BC,(1+B)AC,-B(1+A)(1+C),-A(1+B)(1+C),(1+A)(1+B)(1+C))$. To verify the reductions, multiply using $(f\sigma )(g\eta )=f\sigma (g)\sigma \eta$, $s(A)=-A$, $s(B)=C$, $t(B)=-B$, $t(A)=C$, and $AB=C(A+B)$. For example the raw $s$ coefficient in the first word is $A(1+A)(B-C)=(1+A)BC$; its raw identity coefficient is $-(1-A^2)C-A^{2}B=-C-ABC$. Interchanging $A,B$ and $s,t$ gives the second word's identical six coefficients.

All five reverse braid cases. Step 14.1 is a rational identity in independent spectral variables, before assigning colors. Substitute $(x_1,x_2,x_3)=(a+u,b+v,c+w)$ on each source component and apply the color-permuting operators. It proves the full Hecke braid separately on $(a,b,c)$ with all distinct, $(a,a,b)$, $(a,b,b)$, $(a,b,a)$ and $(a,a,a)$, because each is the restriction of the same six-coefficient identity and every $S_r$ preserves ${\mathcal{B}}_{\beta }$ by step 12.1. Even when a displayed spectral denominator has zero constant, cancellation occurs in the original regular operators, and injectivity into fractions preserves the equality. Thus the all-equal reverse case also has a complete coefficient computation.

Faithfulness for the Hecke action. The mixed relation moves polynomials to the right of $s$-words and the exact symmetric-group relations reduce each word to $s_w$, so $H_n^{\mathrm{d}\mathrm{g}}$ is spanned by $s_{w}f(x)$. In the action of steps 12.1–15.1 its leading permutation coefficient is the product of nonzero rational factors $1+Z^{-1}$ along a reduced word. All lower terms have shorter permutation length. The shortest-relation argument of step 10.1 proves independence and therefore the affine Hecke PBW basis. Completing at the central ideal of step 1.1 gives the finite coefficientwise PBW module ${\sum }_{w,\nu }{s}_{w}k[\text{\hspace{0.17em}⁣}[y]\text{\hspace{0.17em}⁣}]e(\nu )$; its operator action remains faithful by the same argument over formal fractions. The coefficients $1+Z^{-1}$ need only be nonzero in that field, not units in the coefficient ring.

Comparing the explicit tables. On an unequal source $(a,b)$, $S_r+p_{ab}$ has only the crossing term. Its target coefficient is $1+{\zeta }^{-1}$, where $\zeta =b-a+u-v$. Right multiplication by $q_{ab}^{-1}$ substitutes $(v,u)$ on that target. If $a\to b$, $q_{ab}(v,u)=1+{\zeta }^{-1}$, giving coefficient $1$; if $a\leftarrow b$, $q_{ab}(v,u)=-{\zeta }^{-1}$ and $\zeta =-1+u-v$, giving coefficient $-(\zeta +1)=v-u$; off edges the quotient is again $1$. On equal colors $S_r={\sigma }_r-(1-{\sigma }_r)\mathrm{/}(u-v)$, and $(S_r+1)(1+v-u{)}^{-1}=({\sigma }_r-1)\mathrm{/}(u-v)=E_r$ by moving the inverse through ${\sigma }_r$. Thus $(S_r+p)q^{-1}={\psi }_r$ in the faithful action. Equivalently ${\psi }_{r}q-p=S_r$.

Homomorphisms in both directions. Faithfulness in steps 10.1–11.1 and 16.1 transfers the concrete equalities to the completed presented algebras. Step 17.1 identifies the forward images with the operators checked in steps 4.1–9.1, so all KLR relations hold in ${\widehat{H}}_{\beta }^{\mathrm{d}\mathrm{g}}$. It identifies the reverse images with the operators checked in steps 13.1–15.1, so all Hecke relations hold in ${\widehat{R}}_{\beta }$. The coefficient-algebra maps identify the central completion topologies: the symmetric ideal vanishing at the spectral multiset and the all-zero dot ideal have cofinal powers on ${\mathcal{B}}_{\beta }$ by step 1.1. Therefore the maps extend continuously to the stated completions. This proves both directions locally rather than importing a cyclic finite-$e$ theorem.

Inverse composites and uniformity. The coefficient maps fix every $e(\nu )$ and invert $y_r=x_r-{\nu }_r$ componentwise. The crossing composites are $({\psi }_{r}q-p+p)q^{-1}={\psi }_r$ and $((s_r+p)q^{-1})q-p=s_r$, with the same right multiplication order. Hence each composite is the identity on its generators, coefficient series and completion. None of $p,q,c$ depends on the rank, multiset, interval endpoints, level or cyclotomic charge, so enlarging an interval preserves every formula on the old tuples. Nilpotent dots permit finite evaluation of series, and every proved identity survives a quotient. Rank zero has coefficient algebra $k$ and both maps are the identity; empty positive-rank sectors have zero algebra. No assertion about cyclotomic support, based Grothendieck classes or Specht modules is needed for this local comparison.