Cyclotomic dot annihilators, finite-dimensional blocks, and finite root strings

The vacuum and the first-strand annihilator. At $\beta =0$, the sole word is empty and $R^a(0)=k$, since there is no first strand and no cyclotomic relation to impose. This block is finite dimensional of dimension $1$, and its no-dot annihilator and nilpotence assertions are vacuous. The first-strand computation now concerns only nonempty words in positive-height blocks. Let $g_1:={\prod }_{j\in I}{a}_j\in k[u]$, monic of degree $\mathrm{\Lambda }:={\sum }_{j\in I}{\lambda }_j$. For every nonempty $\nu$ and every positive-height block, $a_{{\nu }_1}(x_1)e(\nu )=0$ by definition of the quotient, and $g_1$ is divisible by $a_{{\nu }_1}$, so $g_1(x_1)e(\nu )=0$ for all such $\nu$: the monic polynomial $g_1$ with scalar coefficients annihilates the first dot $x_1$ in every positive-height block, with the convention that a zero corner satisfies the identity trivially.

Equal-colour exchange. Suppose a one-variable polynomial $f$ satisfies $f(x_a)e(\nu )=0$ on a block with ${\nu }_a={\nu }_{a+1}=j$; write $x=x_a$, $y=x_{a+1}$ and put ${\kappa }_a:=-{\tau }_a$. Then ${\kappa }_a^2={\tau }_a^2=Q_{jj}(x,y)=0$ because $Q_{jj}:=0$ (Q1), and from the equal-colour dot moves ${\kappa }_{a}x=y{\kappa }_a-e$ and ${\kappa }_{a}y=x{\kappa }_a+e$ one computes ${\kappa }_{a}f(x)=f(y){\kappa }_a+\mathrm{\partial }f$ and ${\kappa }_{a}f(x){\kappa }_a=f(y){\kappa }_a^2+\mathrm{\partial }f{\kappa }_a=\mathrm{\partial }f{\kappa }_a$, where $\mathrm{\partial }f:=(f(y)-f(x))\mathrm{/}(x-y)\in k[x,y]$ is the divided difference and $f(y)$ is the same one-variable polynomial evaluated at $y$. Applying the identity $(x-y){\kappa }_{a}f{\kappa }_{a}e(\nu )=(s_{a}f-f){\kappa }_{a}e(\nu )=({\kappa }_{a}f-\mathrm{\partial }f-f{\kappa }_a)e(\nu )$ to a module on which $f(x)e(\nu )$ acts as zero, the terms containing $f$ vanish and give $\mathrm{\partial }fe(\nu )M=0$, whence $f(y)e(\nu )M=(f(x)+(x-y)\mathrm{\partial }f)e(\nu )M=0$. Hence $f(x_{a+1})e(\nu )$ also annihilates the module, where $\mathrm{\partial }f=(s_{a}f-f)\mathrm{/}(x-y)$ is the divided difference of [F3] with the sign convention of the polynomial representation. For any multivariate polynomial $H$ in the commuting dots on an equal-colour corner, the same relations give ${\kappa }_{a}H=(s_{a}H){\kappa }_a+{\mathrm{\partial }}_{a}H$. If $He(\nu )=0$, then $(x_a-x_{a+1}){\kappa }_{a}H{\kappa }_{a}e(\nu )=({\kappa }_{a}H-{\mathrm{\partial }}_{a}H-H{\kappa }_a)e(\nu )$ implies ${\mathrm{\partial }}_{a}He(\nu )=0$; multiplying the defining divided-difference equation also gives $(s_{a}H)e(\nu )=0$. Thus this closure applies to each polynomial produced by the iterated differences, not just the initial one-variable annihilator.

Unequal-colour exchange. Let $f\in k[u]$ be monic of degree $d$ and suppose $f(x_a)e({\nu }^{\mathrm{\prime }})=0$ for every word ${\nu }^{\mathrm{\prime }}$ in the current block; this is the global position annihilator supplied by the induction below. Fix $\nu$ with ${\nu }_a=j\mathrm{\ne }l={\nu }_{a+1}$ and write $x=x_a$, $y=x_{a+1}$. Since $s_a\nu$ is another word in that same block, $f(x)e(s_a\nu )=0$, and the dot slides give ${\tau }_{a}f(x){\tau }_{a}e(\nu )=f(y)Q_{jl}(x,y)e(\nu )=0$. Put $m=-a_{lj}$ and let $c_y$ be the coefficient of $y^m$ in $Q_{jl}(x,y)$. By (Q3), positivity of the dot degrees excludes all higher $y$-powers and any $x$ factor in that coefficient, while (Q4) makes $c_y\in k^{\times }$; thus $Q_{jl}(x,y)=c_y{y}^m+{\sum }_{b<m}{q}_b(x)y^b$. The algebra $k[y][x]\mathrm{/}(f(x))$ is free of rank $d$ on $1,x,\dots ,x^{d-1}$. Multiplication by $Q_{jl}$ has matrix $c_y{y}^{m}I+{\sum }_{b<m}{T}_b{y}^b$, so its determinant $\rho (y)={\mathrm{Res}}_x(f(x),Q_{jl}(x,y))$ has degree $dm$ and leading coefficient $c_y^d$. The adjugate identity, applied to $1$ in that free algebra, gives a polynomial $B(x,y)$ with $Q_{jl}B\equiv \rho (\mathrm{m}\mathrm{o}\mathrm{d}f)$, hence $\rho =Af(x)+B{Q}_{jl}$ for a polynomial $A(x,y)$. Therefore $h(y):=c_y^{-d}f(y)\rho (y)$ is monic of degree $d(1-a_{lj})$ and annihilates the $\nu$ corner: both $f(x)f(y)$ and $f(y)Q_{jl}$ in its displayed polynomial-ideal expression vanish there. If $d=0$ the block is already zero; if $m=0$, (Q3) makes $Q_{jl}=c_y$ and $f(y)$ itself annihilates that corner. Only the actual nonzero extremal scalar is inverted, so the argument works over every stated field.

Global monic position annihilators. Step 1.1 supplies a single monic $g_1$ with scalar coefficients annihilating $x_1$ on every word corner. Suppose a monic $g_a$ annihilates $x_a$ on all word corners of a fixed block. Across an equal-colour pair, step 1.2 shows that $g_a(x_{a+1})$ vanishes; across an unequal pair $(j,l)$, step 1.3 applies because $g_a$ also vanishes on the adjacent swapped word, and supplies a monic $h_{jl}$ in the next dot. The finite product $g_{a+1}:=g_a{\prod }_{j\mathrm{\ne }l}{h}_{jl}$ is monic with scalar coefficients and annihilates $x_{a+1}$ on every corner, whichever adjacent pair occurs. This is a constructive finite product over the finite colour set, and the same formulas can be used simultaneously for every block of the given length. The resulting degree bounds depend on the fixed position and length, as well as the Cartan datum and the initial annihilator degree. Induction gives genuine global position annihilators; a finite product over positions also gives one annihilator for every dot of a fixed block.

Finite-dimensionality. Fix $\beta$ with $\mathrm{\mid }\beta \mathrm{\mid }=n$ and let $g_a$ be a monic annihilator of $x_a$ on $R^a(\beta )$ as in step 2.1, of degree $d_a$. By the PBW basis [F2], each corner $e(\nu )R^a(\beta )e({\nu }^{\mathrm{\prime }})$ is spanned by the classes of ${\tau }_w{x}_1^{m_1}\cdots x_n^{m_n}e(\nu )$; using $g_a(x_a)=0$, any exponent $m_a\ge d_a$ is reduced to a $k$-linear combination of smaller powers of $x_a$ (the $x$'s commute), so every element of the corner is a combination of the finitely many classes with $0\le m_a<d_a$. Since $I^{\beta }$ is finite and $S_n$ is finite, $R^a(\beta )$ is finite dimensional over $k$.

Nilpotence in the standard homogeneous case. Suppose $a_j(u)=u^{{\mu }_j}$ for monic powers, so $x_1$ is killed by $x_1^{\mathrm{\Lambda }}$ with $\mathrm{\Lambda }={\max }_j{\mu }_j$. If $x_a^N=0$ on a block and the colours are equal, the computation of step 1.2 with $f(u)=u^N$ gives $x_{a+1}^N=0$ (both $f(x)$ and $\mathrm{\partial }f=\frac{y^N-x^N}{x-y}=-{\sum }_{r=0}^{N-1}{x}^{N-1-r}{y}^r$ kill the module). If the colours differ $j\mathrm{\ne }l$, write $Q_{jl}(x,y)=c{y}^q+xH(x,y)$ with $q:=-a_{lj}\ge 0$ and $c\mathrm{\ne }0$, using (Q3)-(Q4); the identity ${\kappa }_a{x}^N{\kappa }_a=y^N{Q}_{jl}(x,y)$ gives $c{y}^{N+q}=-x{y}^{N}H(x,y)$ on the module, and raising this identity of commuting polynomials in $x,y$ to the $N$th power kills the right side ($x^N=0$), so $y^{N(N+q)}=0$. Starting from $x_1^{\mathrm{\Lambda }}=0$ and iterating over positions, every dot $x_a$ is nilpotent, with an exponent obtained by the finitely many displayed transformations.

All-$i$ tails vanish. Fix a prefix length $\mathrm{\ell }$. For each of the finitely many prefixes $w\in I^{\mathrm{\ell }}$, choose the explicit scalar monic position annihilator $g_w$ for $x_{\mathrm{\ell }+1}$ on the block $wi$ supplied by step 2.1, of degree $d_w$, and put $d={\max }_w{d}_w$. On the word $w{i}^{d_w+1}$, the first-$\mathrm{\ell }+1$-strand embedding carries the cyclotomic relations and $g_w(x_{\mathrm{\ell }+1})e(wi)$ into the same relations in $R^a(\mathrm{\ell }+d_w+1)$, so that polynomial vanishes on the longer corner. The general equal-colour polynomial calculation of step 1.2 then shows successively that ${\mathrm{\partial }}_{\mathrm{\ell }+d_w}\cdots {\mathrm{\partial }}_{\mathrm{\ell }+1}{g}_w(x_{\mathrm{\ell }+1})$ vanishes there. For the sign convention ${\mathrm{\partial }}_a=(s_a-1)\mathrm{/}(x_a-x_{a+1})$, direct subtraction of the generating functions ${\prod }_j(1-z_{j}t{)}^{-1}$ gives ${\mathrm{\partial }}_r\cdots {\mathrm{\partial }}_1{z}_1^s=(-1{)}^r{h}_{s-r}(z_1,\dots ,z_{r+1})$, where $h_t$ is the complete homogeneous polynomial, $h_0=1$ and $h_t=0$ for $t<0$. Applied with $r=d_w$, this length-$d_w$ chain sends the monic leading term to $(-1{)}^{d_w}$ and every lower term to $0$. Thus the vanishing polynomial is the unit $(-1{)}^{d_w}e(w{i}^{d_w+1})$, and that word corner is zero. No full longest-permutation alternator is substituted for this shorter chain. Further suffix extension shows every word with $d+1$ consecutive $i$'s starting at position $\mathrm{\ell }+1$ is zero; the case $d_w=0$ already has $g_w=1$ and the corner zero.

Finite $i$-strings. Fix $i$. For a word $w$ over $I$ let ${\mathrm{\#}}_i(w)$ be the number of letters equal to $i$. Claim: for every $b\ge 0$ there is an integer $L_b$ such that every word with at most $b$ letters different from $i$ and $\mathrm{\mid }w\mathrm{\mid }\ge b+L_b$ is zero in the quotient. For $b=0$: an all-$i$ word of length $\ge L_0:=d+1$, where $d$ is a uniform annihilator degree at prefix length $0$ from step 2.1, is zero by step 3.3. Assume $L_{b-1}$; put $k:=L_{b-1}$, $\mathrm{\ell }:=b-1+k$, and let $m$ be an all-$i$ tail bound at prefix length $\mathrm{\ell }$ obtained from step 3.3 (a finite maximum over the finitely many prefixes of length $\mathrm{\ell }$). Set $L_b:=k+m-1$ and let $w$ have at most $b$ non-$i$ letters and $\mathrm{\mid }w\mathrm{\mid }\ge b+L_b=\mathrm{\ell }+m$. If the prefix of length $\mathrm{\ell }$ contains at least $k$ letters $i$ it has at most $b-1$ non-$i$ letters and length $\mathrm{\ell }=(b-1)+L_{b-1}$, so it is zero by the induction hypothesis and $w=0$. Otherwise the prefix contains at most $k-1$ letters $i$ and therefore at least $\mathrm{\ell }-(k-1)=b$ non-$i$ letters. It already contains all the at most $b$ non-$i$ letters of $w$, so every remaining letter is $i$; there are at least $m$ remaining positions, and the tail is zero by step 3.3; in both cases $w=0$.

Conclusion. Fix $\beta$ and put $b={\sum }_{j\mathrm{\ne }i}{\beta }_j$. For every integer $m\ge \max (0,L_b-{\beta }_i)$, a word of weight $\beta +m{\alpha }_i$ has exactly $b$ non-$i$ letters and length $b+{\beta }_i+m\ge b+L_b$, so step 4.1 makes every such word corner zero. Hence $R^a(\beta +m{\alpha }_i)=0$ for all sufficiently large $m$, giving finite right-append $i$-strings. Step 3.1 gives finite dimensionality of every block, and step 3.2 gives nilpotence of every dot in the standard homogeneous specialization. The monic inhomogeneous branch is ungraded throughout. The global annihilator, determinant normalization and tail argument use only finite polynomial computations and units already provided by (Q4), with no new hypotheses.