2.1 Iwahori factorization

Let p be a prime. For the sake of clarity, we will discuss everything with the assumption that our ground field is ${{\mathbb Q}_p}$ and just note at the beginning that most of the results in this preliminary section are usually stated in their corresponding sources in an appropriately generalized fashion for a finite extension $L/{{\mathbb Q}_p}$ .

We fix a simple, simply connected reductive group $\mathbf {{G}}$ which is split over ${{\mathbb Q}_p}$ , and choose an ${{\mathbb Q}_p}$ -split maximal torus $\mathbf {{T}}\subset \mathbf {{G}}$ . Let $(X^*(\mathbf {{T}}), \Phi , X_*(\mathbf {{T}}), \Phi ^{\vee })$ be the associated root datum. We fix a system of positive roots $\Phi ^+$ with simple roots $\Delta \subset \Phi ^+$ and let $\Phi ^-=-\Phi ^+$ . Recall than any root in $\Phi $ can be uniquely written as a linear combination of simple roots and the sum of the coefficients gives us the height function $\text {ht}: \Phi \rightarrow {\mathbb Z}$ on the roots. We recall that the Coxeter number of $\mathbf {{G}}$ is ${h=1+\text {ht}(\theta )}$ where $\theta $ is the highest root.

Let $G=\mathbf {{G}}({{\mathbb Q}_p})$ with the usual locally profinite topology, and similarly $T=\mathbf {{T}}({{\mathbb Q}_p})$ . Further we will denote the maximal compact subgroup by $T^0\subset T$ and its Sylow pro-p-subgroup by $T^1$ . Recall that to each $\alpha \in \Phi $ one can attach a unipotent root subgroup $\mathbf {{U}}_{\alpha }$ normalized by $\mathbf {{T}}$ . We let $U_{\alpha }=\mathbf {{U}}_{\alpha }({{\mathbb Q}_p})$ and choose a natural isomorphism $x_{\alpha }: {{\mathbb Q}_p} \overset {\sim }{\longrightarrow } U_{\alpha }$ such that $tx_{\alpha }(x)t^{-1}=x_{\alpha }(\alpha (t)x)$ for all $t \in T$ and $x \in {{\mathbb Q}_p}$ . This gives a filtration of $U_{\alpha }$ by the subgroups $U_{\alpha ,r}=x_{\alpha }((p^r))$ for varying $r \in {\mathbb Z}$ .

We consider the full Iwahori subgroup J corresponding to $\Phi ^+$ . We don’t pursue the geometric aspects of J and instead just note that the so-called Iwahori factorization says that multiplication defines a homeomorphism,

$$ \begin{align*}\prod_{\alpha\in \Phi^-}U_{\alpha,1} \times T^0 \times \prod_{\alpha\in \Phi^+}U_{\alpha,0} \overset{\sim}{\longrightarrow} J. \end{align*} $$

We concentrate our attention on the Sylow pro-p subgroup of J, namely the pro-p Iwahori subgroup I. Again multiplication gives a homeomorphism,

(2.1.0) $$ \begin{align} \prod_{\alpha\in \Phi^-}U_{\alpha,1} \times T^1 \times \prod_{\alpha\in \Phi^+}U_{\alpha,0} \overset{\sim}{\longrightarrow} I. \end{align} $$

Here the products over $\Phi ^-$ and $\Phi ^+$ are ordered in an arbitrarily chosen way, see the proof of [Reference Ollivier and SchneiderOS19, Lemma 2.3] and [Reference Ollivier and SchneiderOS19]. From Section 2.4, we will fix an ordering of this product to discuss ordered basis and subsequent developments that depend on the ordered basis.

2.2 p-saturated groups

To make the article somewhat self-contained, we collect here a few definitions related to p-valuation on a group H which can be any pro-p, torsion free, p-adic Lie group. Let us recall that a p -valuation [Reference SchneiderSch11, Chapter 23, p. 169] $\omega $ on the group H is a real valued function, $\omega : H \setminus \{1\} \rightarrow (\frac {1}{p-1},\infty )$ with the convention $\omega (1) = \infty $ , satisfying

1. (1) $\omega (g^{-1}h) \ge \min (\omega (g),\omega (h))$ ,

2. (2) $\omega ([g,h])\ge \omega (g)+ \omega (h)$ ,

3. (3) $\omega (g^p)= \omega (g)+1$

for $g,h \in H$ .

A sequence of elements $(g_1, \cdots , g_d)$ in a group H equipped with a p-valuation $\omega $ is called an ordered basis [Reference SchneiderSch11, Definition, Chapter 26, p. 182] of $(H, \omega )$ if it satisfies

1. (1) $\mathbb {Z}_p^d \rightarrow H, (x_1,\cdots , x_d) \rightarrow g_1^{x_1}\cdots g_d^{x_d}$ is a homeomorphism,

2. (2) $\omega (g_1^{x_1}\cdots g_d^{x_d})= \min _{1\le i\le d} (\omega (g_i)+v_p(x_i))$ for any $x_1,\cdots , x_d\in {\mathbb Z_p}$ .

Here $v_p(-)$ is the (normalized) p-adic valuation on ${\mathbb Z}_p$ with $v_p(p)=1$ . Finally a p-valuable group $(H,\omega )$ equipped with the valuation $\omega $ is saturated [Reference SchneiderSch11, Definition, Chapter 26, p. 187] if any g with $\omega (g)> \frac {p}{p-1}$ is a p-th power of some element of H.

2.3 Additive ordering on the set of positive roots

We recollect results from [Reference PapiPap94] in a form we will need to prove Lemma 5.3.1 later.

Let W be the associated Weyl group to the root system $\Phi $ . Let $\Delta =\{\delta _1,\cdots , \delta _n\}$ let $s_1,\cdots , s_n$ be the reflections corresponding to simple roots.

In particular note that, if $w_0$ is the longest element of the Weyl group then $\Phi ^{+}$ is associated with $w_0$ .

The main result of [Reference PapiPap94] is that

Since we already observed that $\Phi ^+$ is associated with $w_0$ , as a corollary of 2.3.3 we get

2.4 Ordered basis of pro-p Iwahori

We recall what we need from [Reference Lahiri and SorensenLS22, Section 3] with slight modification and without any proofs.

In [Reference Lahiri and SorensenLS22] it was shown that the pro-p Iwahori subgroup I can be equipped with a p-valuation $\omega $ with respect to which I is saturated. For us it will be important to work with a choice of an ordered basis for $(I, \omega )$ and to be able to explicitly write down the valuations of these basis elements.

Let us fix any ordering on the elements of $\Phi ^{-}$ with increasing height function on the roots, the elements of $\Delta $ in an arbitrary way and the elements of $\Phi ^+$ with respect to an arbitrary but fixed compatible ordering in the sense of Definition 2.3.2. This gives a total ordering on the roots starting from the negative roots, then the simple roots and finally the positive roots.

Theorem 2.4.1 ([Reference Lahiri and SorensenLS22, Proposition 3.4, Corollary 3.6])

With $p-1> h,$ an ordered basis with their respective valuations for $(I, \omega )$ is given by

$$\begin{align*}\begin{array}{ccccccccccc} &\bullet& &x_\beta(p)_{\beta\in \Phi^{-}}&, &\omega(x_\beta(p)) &=& 1+\frac{\mathrm{ht}(\beta)}{h},&\\ &\bullet& &\delta^\vee(1+p)_{\delta \in \Delta}&, &\omega(\delta^\vee(1+p))&=& 1,&\\ &\bullet& &x_\alpha(1)_{\alpha\in \Phi^+},&, &\omega(x_\alpha(1))&=& \frac{\mathrm{ht}(\alpha)}{h}.& \end{array}\end{align*}$$

where we order the elements of the ordered basis corresponding to a total ordering of $\Phi $ chosen as above.

Note that the description given in [Reference Lahiri and SorensenLS22] chooses $e^p$ as an ordered basis of $1+ (p)$ . Here to make the calculations more straight forward the natural choice seems to be $1+p$ . From this point onward we change our notation to write $h_\delta $ instead of $\delta ^\vee $ to match with the notation used in [Reference RayRay20].

This particular choice of an ordered basis will be used crucially in the computations presented in Section 5.3.

2.5 Iwasawa Algebra

Let H be any profinite group and let ${\mathcal N}(H)$ the set of open normal subgroups in H. Then we know that

$$\begin{align*}H = \varprojlim_{N\in {\mathcal N}(H)}H/N\end{align*}$$

is the projective limit, as a topological group, of the finite groups $H/N$ equipped with the discrete topology. By functoriality of associated group rings, the algebraic group rings ${\mathbb Z}_p[H/N]$ form, for varying N in ${\mathcal N}(H)$ , a projective system of rings. The projective limit

$$\begin{align*}\Lambda(H) := {\mathbb Z}_p[[H]] := \varprojlim_{N\in {\mathcal N}(H)}{\mathbb Z}_p[H/N]\end{align*}$$

is called the completed group ring or the Iwasawa algebra of H over ${\mathbb Z}_p$ . Equipping $\Lambda (H)$ with the projective limit topology of the natural topologies of ${\mathbb Z}_p[H/N]$ makes it a complete pseudo-compact topological ring.

The embedding of $H \rightarrow \Lambda (H)$ promotes to an injective embedding of the group ring ${\mathbb Z}_p[H]$ into $\Lambda (H)$ with dense image.

Now let $(H, \omega )$ be a p-saturated group with a p-valuation $\omega $ and a choice of ordered basis $(h_1,\cdots , h_d)$ . In this situation, we get a homeomorphism of analytic manifolds

$$ \begin{align*} c:\mathbb{Z}_p^d &\rightarrow H \\ (x_1,\cdots, x_d) &\mapsto h_1^{x_1}\cdots h_d^{x_d}. \end{align*} $$

We use the standard notation $b_j= h_j-1\in \Lambda (H)$ , and for $\alpha =(\alpha _1,\cdots ,\alpha _d)\in \mathbf {N}^d$ we let $\mathbf {b}^\alpha = b_1^{\alpha _1}\cdots b_d^{\alpha _d}$ . Then any element $\lambda \in \Lambda (H)$ can be written as $\lambda := \sum _\alpha c_\alpha \mathbf {b}^\alpha $ with $c_\alpha \in {\mathbb Z}_p$ and we can naturally equip it with the valuation,

(2.5.0) $$ \begin{align} \tilde{\omega}_\Lambda(\sum_\alpha c_\alpha \mathbf{b}^\alpha)= \inf_\alpha(v_p(c_\alpha)+\sum_{i=1}^d \alpha_i\omega(h_i) ). \end{align} $$

Since $\Lambda (H)$ can be viewed as the ${\mathbb Z}_p$ -valued continuous distributions on H (cf. [Reference SchneiderSch11, Lemma 21.1]), after pulling back and dualizing we get an isomorphism of topological ${\mathbb Z_p}$ -modules

$$ \begin{align*} c_\star: \Lambda(\mathbb{Z}_p^d) &\cong \Lambda(H). \end{align*} $$

By [Reference SchneiderSch11, 20.1] we know that $\Lambda (\mathbb {Z}_p^d) \cong {\mathbb Z_p}[[X_1,\cdots , X_d]]$ the power series ring in d variables, as a topological ring.

Using Theorem 2.4.1 we thus rewrite for I,

Lemma 2.5.1 (Lazard)

The following are isomorphic as ${\mathbb Z_p}$ -modules with an isomorphism given by,

$$ \begin{align*} \tilde{c}: {\mathbb Z_p}[[U_\beta, V_\alpha, W_\delta: \beta\in \Phi^{-}, \alpha\in \Phi^{+}, \delta\in \Delta]] \cong \Lambda(I)\\ 1+ V_\alpha \mapsto x_\alpha(1),\\ 1+ W_\delta \mapsto h_\delta(1+p),\\ 1+ U_\beta \mapsto x_\beta(p). \end{align*} $$

2.6 The graded structure

For any p-valued group H, the valuation $\tilde {\omega }_{\Lambda }$ induces a natural filtration of $\Lambda (H)$ as follows. Define for $v \ge 0$ ,

$$\begin{align*}\begin{array}{cccc} \Lambda(H)_v &:=\big\{\lambda\in \Lambda(H)\big | \tilde{\omega}_{\Lambda}(\lambda)\ge v \big\}, \\ \Lambda(H)_{v^+} &:=\big\{\lambda\in \Lambda(H)\big | \tilde{\omega}_{\Lambda}(\lambda)> v \big\}. \end{array}\end{align*}$$

This gives a filtration and the associated graded algebra

$$\begin{align*}\mathrm{gr}^v\Lambda(H):= \Lambda(H)_v/\Lambda(H)_{v^+}; \hspace{.2cm} \mathrm{gr}\Lambda(H)=\bigoplus_{v\ge 0}\mathrm{gr}^v\Lambda(H). \end{align*}$$

Under this filtration $\Lambda (I)$ is filtered by $\frac {1}{h}{\mathbb N}$ .

This follows from the fact that, $\mathrm {gr}^v\Lambda (I)$ is generated by the independent elements $p^s\prod _{\beta \in \Phi ^{-}} U_\beta ^{n_\beta }\prod _{\delta \in \Delta } W_\delta ^{n_\delta }\prod _{\alpha \in \Phi ^{+}}V_{\alpha }^{n_\alpha }$ such that

$$\begin{align*}s+ \sum_{\beta\in \Phi^{-}} \frac{h+{\mathrm{ht}(\beta)}}{h}n_\beta+ \sum_{\delta\in \Delta} n_\delta+\sum_{\alpha\in \Phi^{+}}\frac{{\mathrm{ht}(\alpha)}}{h}n_\alpha= v.\end{align*}$$

The filtration on $\Lambda (I)$ induces a natural filtration and thus a grading on $\Omega (I)= \Lambda (I)\otimes _{\mathbb Z_p} {{\mathbb F}_p}$ . We will use this to compute the dimension of $\mathrm {gr}^v\Omega (I)$ as an $\mathbb {F}_p$ -vector space. We know that $\mathrm {gr}^v\Omega (I)$ is generated by $\prod _{\beta \in \Phi ^{-}} U_\beta ^{n_\beta }\prod _{\delta \in \Delta } W_\delta ^{n_\delta }\prod _{\alpha \in \Phi ^{+}}V_{\alpha }^{n_\alpha }$ with

$$\begin{align*}\sum_{\beta\in \Phi^{-}} \frac{h+{\mathrm{ht}(\beta)}}{h}n_\beta+ \sum_{\delta\in \Delta} n_\delta+\sum_{\alpha\in \Phi^{+}}\frac{{\mathrm{ht}(\alpha)}}{h}n_\alpha= v.\end{align*}$$

We don’t change the filtered pieces if we replace $v\in \frac {1}{h}{\mathbb N}$ by $m= vh$ . This change assigns the following valuations (which we still denote by $\tilde {\omega }_{\Lambda }$ by abuse of notation) to the generators of the Iwasawa algebra.

$$\begin{align*}\begin{array}{ccccccccccc} &\tilde{\omega}_{\Lambda}(U_\beta)&=& h+{\mathrm{ht}(\beta)},&\\ &\tilde{\omega}_{\Lambda}(W_\delta)&=& h,&\\ &\tilde{\omega}_{\Lambda}(V_\alpha)&=& {\mathrm{ht}(\alpha)}.& \end{array}\end{align*}$$

The remarks above recover the following fact which is really due to Lazard.

3.1 The Chevalley relations

The Chevalley relations on the Chevalley basis of I translate to relations between the non-commutative variables defining the Iwasawa algebra. As remarked in the introduction the main result of this article is that these are a defining set of relations for the Iwasawa algebra.

In $\mathbf {G}({\mathbb Q}_p)$ , we have the following Chevalley relations (see [Reference SteinbergSte16, p. 23]).

(Diag) $h_\alpha (u) h_\beta (t)= h_\beta (t) h_\alpha (u).$ Here $\alpha ,\beta \in \Delta .$

(Diag-uni) $h_\delta (t) x_\alpha (u) h_\delta (t)^{-1}= x_\alpha (t^{\langle \alpha , \delta \rangle } u)$ where $\langle \alpha , \delta \rangle \in {\mathbb Z}$ are the Cartan integers as defined in [Reference SteinbergSte16, p. 2, point (2)]. Here $\alpha \in \Phi $ and $\delta \in \Delta $ .

(Uni-1) If $\alpha +\beta \neq 0$ and $\alpha +\beta \notin \Phi $ , then we have $x_{\alpha }(t) x_{\beta }(u)= x_{\beta }(u)x_{\alpha }(t).$

(Uni-2) If $\alpha +\beta \neq 0$ and $\alpha +\beta \in \Phi $ , then we have $x_\alpha (t)x_\beta (u)= \prod _{\underset {i\alpha + j\beta \in \Phi }{i, j> 0,}} x_{i\alpha + j\beta }(c_{ij} t^i u^j)x_\beta (u)x_\alpha (t),$ where $c_{ij}$ ’s are unique integers depending on $\alpha $ and  $\beta $ .

(Uni-3) Let $\alpha \in \Phi ^+$ . Writing $\alpha =\sum _{\delta _i\in \Delta }n_i\delta _i$ , we have $h_\alpha (t)= \prod _{\delta _i\in \Delta }h_{\delta _i}(t)^{n_i}$ for some non-negative integers $n_i$ . We obtain

$$ \begin{align*}x_\alpha(1) x_{-\alpha}(p)= x_{-\alpha}(p)^Q\prod_{\delta_i\in \Delta}h_{\delta_i}(1+p)^{n_i}x_{\alpha}(1)^Q,\end{align*} $$

where $Q=(1+p)^{-1}.$ Here the variables $u,t$ are from ${\mathbb Q}_p$ if they are in the argument of $x_{*}(-)$ and from ${\mathbb Q}_p^\times $ if they are in the argument of $h_{*}(-).$ Also note that the expressions $x_{\alpha }(1)^Q$ and $x_{-\alpha }(p)^Q$ are well-defined (cf. [Reference Dixon, du Sautoy, Mann and SegalDdSMS99, Section 1.3]).

We include a proof of (Uni-3); the other relations are exactly as in [Reference SteinbergSte16]. For any root $\alpha \in \Phi ^+$ , we have a homomorphism

$$ \begin{align*}\phi_\alpha: SL_2({\mathbb Q}_p) \rightarrow \langle x_{\alpha}(t), x_{-\alpha}(t); t \in {\mathbb Q}_p\rangle\end{align*} $$

such that

$$ \begin{align*} \phi_\alpha(1+tE_{12})&=x_\alpha(t),\\ \phi_\alpha(1+tE_{21})&=x_{-\alpha}(t),\\ \phi_{\alpha}(tE_{11}+t^{-1}E_{22})&=h_\alpha(t), \end{align*} $$

where $E_{ij}$ is the $2 \times 2$ matrix with 1 in the $(i,j)$ -th entry and zero elsewhere. In $SL_2({\mathbb Z}_p)$ we have the following matrix relation

$$ \begin{align*}(1+E_{12})(1+pE_{21})=\big(1+p(1+p)^{-1}E_{21}\big)\big((1+p)E_{11}+(1+p)^{-1}E_{22}\big)\big(1+(1+p)^{-1}E_{12}\big).\end{align*} $$

Since $\phi _{\alpha }$ is a homomorphism we obtain

$$ \begin{align*}x_{\alpha}(1)x_{-\alpha}(p)=x_{-\alpha}(p(1+p)^{-1})h_{\alpha}(1+p)x_\alpha((1+p)^{-1}),\end{align*} $$

which implies

$$ \begin{align*}x_{\alpha}(1)x_{-\alpha}(p)= x_{-\alpha}(p)^{(1+p)^{-1}}\prod_{\delta_i\in \Delta}h_{\delta_i}(1+p)^{n_i}x_{\alpha}(1)^{(1+p)^{-1}}.\end{align*} $$

3.2 Relations in the Iwasawa algebra

The corresponding relations in the Iwasawa algebra are

(R-1) $(1+ W_{\delta _1})(1+ W_{\delta _2})= (1+ W_{\delta _2})(1+ W_{\delta _1}).$ Here $\delta _1, \delta _2 \in \Delta $ .

(R-2) $(1+ W_{\delta })(1+ V_\alpha )= (1+ V_\alpha )^M(1+ W_{\delta }), \alpha \in \Phi ^{+},\delta \in \Delta , M= (1+p)^{\langle \alpha , \delta \rangle }.$

(R-3) $(1+ W_{\delta })(1+ U_\beta )= (1+ U_\beta )^M(1+ W_{\delta }), \beta \in \Phi ^{-}, \delta \in \Delta , M= (1+p)^{\langle \beta , \delta \rangle }.$

(R-4) $V_\alpha U_\beta = U_\beta V_\alpha , \alpha \in \Phi ^{+}, \beta \in \Phi ^{-},\alpha +\beta \neq 0,\alpha +\beta \notin \Phi .$

(R-5) For $\alpha _1, \alpha _2 \in \Phi ^{+}$ , $(1+V_{\alpha _1})(1+V_{\alpha _2})=\underset {\stackrel {i, j>0}{i\alpha _1+j\alpha _2 \in \Phi }}{\prod }(1+V_{i\alpha _1+j\alpha _2})^{c_{ij}}(1+V_{\alpha _2})(1+V_{\alpha _1}).$

(R-6) For $\beta _1, \beta _2 \in \Phi ^{-}$ , $(1+U_{\beta _1})(1+U_{\beta _2})=\underset {\stackrel {i, j>0}{i\beta _1+j\beta _2 \in \Phi }}{\prod }(1+U_{i\beta _1+j\beta _2})^{c_{ij}p^{i+j-1}}(1+U_{\beta _2})(1+U_{\beta _1}).$

(R-7) For $\alpha \in \Phi ^{+}, \beta \in \Phi ^{-}$ , $\alpha +\beta \in \Phi \backslash \{0\}$ ,

$$ \begin{align*}(1+V_{\alpha})(1+U_{\beta})=\underset{\stackrel{i, j>0}{i\alpha+j\beta \in \Phi^{-}}}{\prod}(1+U_{i\alpha+j\beta})^{c_{ij}p^{j-1}}\underset{\stackrel{i, j>0}{i\alpha+j\beta \in \Phi^{+}}}{\prod}(1+V_{i\alpha+j\beta})^{c_{ij}p^j}(1+U_{\beta})(1+V_{\alpha}).\end{align*} $$

(R-8) With $\alpha = \sum _i n_i \delta _i, (1+V_\alpha )(1+ U_{-\alpha })= (1+ U_{-\alpha })^Q\underset {\delta _i\in \Delta }{\prod }(1+ W_{\delta _i})^{n_i }(1+V_\alpha )^Q,$ where $Q= (1+p)^{-1}.$

In the relations above, raising a variable $(1+X)$ to some non-integral power Z i.e., $(1+X)^Z$ will be the usual power series $1+{{Z} \choose {1}}X + {{Z} \choose {2}}X^2 + \cdots .$

5.1 Plan for the proof of Theorem 4.0.3

The theorem is obviously true for $n=0$ as $\mathrm {gr}^0$ is ${{\mathbb F}_p}$ and both sides equal $1$ . For $n=1$ , $\mathrm {gr}^1\overline {{\mathcal B}}$ is a quotient of the space $\mathrm {Fil}^1\overline {{\mathcal A}}/\mathrm {Fil}^{2}\overline {{\mathcal A}}$ with basis $\{V_\alpha , U_\beta \}$ where $\alpha $ varies over all the simple roots (i.e., $\mathrm {ht}(\alpha )=1$ ) and $\beta $ is the negative of the highest root (hence $h+\mathrm {ht}(\beta )=1$ ). Hence

$$ \begin{align*}\dim_{{{\mathbb F}_p}}\mathrm{gr}^1\overline{{\mathcal B}}\le |\Delta|+1=\dim_{{{\mathbb F}_p}} \mathrm{gr}^1\Omega(I).\end{align*} $$

For the general case, we first reduce all the relations $\textbf {(R-1)} -\textbf {(R-8)}$ modulo appropriate filtrations (see Proposition 5.2.1 in Section 5.2). Proposition 5.2.1 tells us how the product of two variables in the wrong order is related to the product of the variables in the right order (the right order is the one prescribed by Theorem 2.4.1). Next we show that we can arrange the variables in the wrong order in $\mathrm {gr}^n\overline {{\mathcal B}}$ , by a sequence of transpositions and put them in the right order, and at each step we reduce the number of inversions (see Lemma 5.3.1 in Section 5.3). This shows that $\dim _{{{\mathbb F}_p}}\mathrm {gr}^n\overline {{\mathcal B}}\le \dim _{{{\mathbb F}_p}} \mathrm {gr}^n\Omega (I)$ since $\mathrm {gr}^n\Omega (I)$ consists of homogeneous symmetric polynomials.

Let us start the proof of Theorem 4.0.3 by analyzing the mod-p reduction of certain binomial coefficients which will be necessary to study the relations between the variables modulo filtrations of an appropriate degree inside $\overline {\mathcal {R}}.$

5.2 Relations modulo appropriate filtrations

Recall that $\deg (W_\delta )= h, \deg (V_\alpha )= \mathrm {ht}(\alpha ), \deg (U_\beta )= h+\mathrm {ht}(\beta )$ .

The variables in relations (R-1) and (R-4) commute modulo any filtration. In this following proposition we compute the other relations modulo appropriate filtrations. We will see that we are reduced to only three relations (R-5’), (R-7’), and (R-8’) given below where the variables don’t commute modulo filtration of an appropriate order.

Note that the powers of the filtrations we are taking in these relations are always the total degree of the variables in the L.H.S plus 1.

Proof Consider relation (R-2)

$$\begin{align*}(1+ W_{\delta})(1+ V_\alpha)= (1+ V_\alpha)^M(1+ W_{\delta}), \alpha\in \Phi^{+}, M= (1+p)^{\langle \alpha, \delta\rangle}. \end{align*}$$

Note that $\deg (W_\delta V_\alpha )=h+\mathrm {ht}(\alpha )$ and so we are interested in studying relation (R-2) modulo $\mathrm {Fil}^{\mathrm {ht}(\alpha )+ h+1}$ .

We want to show that relation (R-2) reduces to

$$\begin{align*}(1+ W_{\delta})(1+ V_\alpha)= (1+ V_\alpha)(1+ W_{\delta}) \text{ } (\mathrm{mod}\text{ } \mathrm{Fil}^{\mathrm{ht}(\alpha)+ h+1}).\end{align*}$$

We will show that by proving

$$\begin{align*}(1+ V_\alpha)^M\equiv (1+ V_\alpha) \text{ } (\mathrm{mod}\text{ } \mathrm{Fil}^{\mathrm{ht}(\alpha)+h+1}). \end{align*}$$

We note that $M\equiv 1 \text { } (\mathrm {mod}\text { } p)$ and

$$\begin{align*}(1+ V_\alpha)^M= 1 + M V_\alpha+ {M\choose 2} V_\alpha^2+\cdots\end{align*}$$

Since degree of $V_\alpha ^m= \mathrm {ht}(\alpha )m$ for any $m\in {\mathbb N}$ we see that $V_\alpha ^m \equiv 0 \text { } (\mathrm {mod}\text { } \mathrm {Fil}^{\mathrm {ht}(\alpha )+h+1})$ as long as $\mathrm {ht}(\alpha )m>\mathrm { ht}(\alpha )+h$ . So we will be done if we can show ${M \choose m} \equiv 0 \text { } (\mathrm {mod}\text { } p)$ for $m\ge 2$ and $\mathrm {ht}(\alpha )m \le \mathrm {ht}(\alpha )+h$ or rewriting, $\mathrm {ht}(\alpha )(m-1) \le h$ then we are done. But note that, if $\mathrm {ht}(\alpha )(m-1)\le h$ then certainly $m-1\le h$ . Thus if we can prove ${M \choose m} \equiv 0 \text { } (\mathrm {mod}\text { } p)$ for $m-1\le h$ we are done. By assumption, $h< p-1$ , thus if $m-1\le h < p-1 $ then $m < p$ . Since $M\equiv 1 \text { } (\mathrm {mod}\text { } p)$ and $2 \leq m <p$ , it is clear that ${M \choose m} \equiv 0 \text { }(\mathrm {mod}\text { } p)$ .

This proves (R-2’).

Next we want to show that relation (R-3)

$$\begin{align*}(1+ W_{\delta})(1+ U_\beta)= (1+ U_\beta)^M(1+ W_{\delta}), \beta\in \Phi^{-}, M= (1+p)^{\langle \beta, \delta\rangle}, \end{align*}$$

reduces to

$$\begin{align*}(1+ W_{\delta})(1+ U_\beta)= (1+ U_\beta)(1+ W_{\delta})\text{ } (\mathrm{mod}\text{ } \mathrm{Fil}^{2h+\mathrm{ht}(\beta)+1}). \end{align*}$$

Just like the last relation we want to show

$$\begin{align*}(1+ U_\beta)^M\equiv 1+ U_\beta \text{ } (\mathrm{mod}\text{ } \mathrm{Fil}^{2h+\mathrm{ht}(\beta)+1}).\end{align*}$$

When $(\mathrm {ht}(\beta )+h)m> 2h+\mathrm {ht}(\beta )$ then $U_\beta ^m \equiv 0 \text { } (\mathrm {mod}\text { } \mathrm {Fil}^{2h+\mathrm {ht}(\beta )+1})$ thus we need to worry about $m\ge 2$ and $(\mathrm {ht}(\beta )+h)m \le 2h+\mathrm {ht}(\beta )$ or rewriting $(\mathrm {ht}(\beta )+h){(m-1)} \le h$ . But pretty much by the same argument as above we are reduced to the case $m-1\le h$ and the above calculation goes through which proves (R-3’).

Next consider relation (R-6). We want to show the relations, for $\beta _1, \beta _2 \in \Phi ^{-}$ ,

$$\begin{align*}(1+U_{\beta_1})(1+U_{\beta_2})=\prod_{\stackrel{i, j>0}{i\beta_1+j\beta_2 \in \Phi}}(1+U_{i\beta_1+j\beta_2})^{c_{ij}p^{i+j-1}}(1+U_{\beta_2})(1+U_{\beta_1}),\end{align*}$$

reduces to

$$\begin{align*}(1+U_{\beta_1})(1+U_{\beta_2})=(1+U_{\beta_2})(1+U_{\beta_1})\text{ } (\mathrm{mod}\text{ } \mathrm{Fil}^{2h+ \mathrm{ht}(\beta_1)+\mathrm{ht}(\beta_2)+1}).\end{align*}$$

We need to analyze $(1+U_{i\beta _1+j\beta _2})^{c_{ij}p^{i+j-1}}$ carefully. As usual

$$ \begin{align*}\deg(U_{i\beta_1+j\beta_2}^m)= m(h+ \mathrm{ht}(i\beta_1+j\beta_2)).\end{align*} $$

Thus for any m such that $m(h+ \mathrm {ht}(i\beta _1+j\beta _2))> 2h+ \mathrm {ht}(\beta _1)+\mathrm {ht}(\beta _2) $ , we have

$$\begin{align*}\deg(U_{i\beta_1+j\beta_2}^m)\equiv 0 \text{ }(\mathrm{mod}\text{ } \mathrm{Fil}^{2h+ \mathrm{ht}(\beta_1)+\mathrm{ht}(\beta_2)+1}).\end{align*}$$

From now on we safely make the assumption,

$$\begin{align*}\begin{array}{ccccc} m(h+ \mathrm{ht}(i\beta_1+j\beta_2)) &\le& 2h+ \mathrm{ht}(\beta_1)+\mathrm{ht}(\beta_2) \\ \text{gives}\; m &\le& \dfrac{2h+ \mathrm{ht}(\beta_1)+\mathrm{ht}(\beta_2)}{h+ \mathrm{ht}(i\beta_1+j\beta_2)}\\ &\le& 2h -2 \;(\text{as } h+ \mathrm{ht}(i\beta_1+j\beta_2)\ge 1, \mathrm{ht}(\beta_k) \leq -1 \text{ for } k=\{1,2\})\\ &\le& 2(p-1)-2= 2p-4 \;(\text{as } p>1+h). \end{array}\end{align*}$$

For any prime $p\ge 3$ and any $k\ge 2$ we have $p^k> 2p- 4$ . Thus, by Lucas’ theorem, we have

$$\begin{align*}{c_{ij}p^{i+j-1}\choose m} \equiv 0 \text{ } (\mathrm{mod}\text{ } p),\; m\le 2p-4 \;\text{and}\;i+j-1 \ge 2.\end{align*}$$

We are left to deal with the case $i+j= 2$ or $i=1, j=1$ . Note that, again by Lucas’ theorem, ${c_{ij}p\choose m} \equiv 0 \text { } (\mathrm {mod}\text { } p)$ for $m < p$ (if $c_{ij}$ is negative, i.e., $c_{ij}=-d_{ij}$ , then ${c_{ij}p\choose m} =(-1)^m {d_{ij}p+m-1\choose m}$ and then apply Lucas’ theorem). Thus we can make the assumption $m \ge p $ . But in this case we have

$$\begin{align*}\begin{array}{ccccccc} m(h+ \mathrm{ht}(\beta_1+\beta_2)) &\ge& p (h+ \mathrm{ht}(\beta_1+\beta_2))\\ &>& (1+h) (h+ \mathrm{ht}(\beta_1+\beta_2)) = h+ \mathrm{ht}(\beta_1+\beta_2)+ h(h+ \mathrm{ht}(\beta_1+\beta_2))\\ &\ge& h+ \mathrm{ht}(\beta_1+\beta_2) + h \text{ }(\text{as } h+ \mathrm{ht}(\beta_1+\beta_2)\ge 1) \\ &=& 2h+ \mathrm{ht}(\beta_1+\beta_2)\\ &=& 2h+ \mathrm{ht}(\beta_1)+\mathrm{ht}(\beta_2). \end{array}\end{align*}$$

Hence this proves (R-6’).

Next consider relation (R-5). We want to show the relation, $\alpha _1, \alpha _2 \in \Phi ^{+}$ ,

$$\begin{align*}(1+V_{\alpha_1})(1+V_{\alpha_2})=\prod_{\stackrel{i, j>0}{i\alpha_1+j\alpha_2 \in \Phi}}(1+V_{i\alpha_1+j\alpha_2})^{c_{ij}}(1+V_{\alpha_2})(1+V_{\alpha_1}),\end{align*}$$

reduces to

$$\begin{align*}V_{\alpha_1}V_{\alpha_2}=\sum_{\stackrel{i, j>0}{i\alpha_1+j\alpha_2 \in \Phi}}c_{ij}V_{i\alpha_1+j\alpha_2}+V_{\alpha_2}V_{\alpha_1}\text{ } (\mathrm{mod}\text{ } \mathrm{Fil}^{\mathrm{ht}(\alpha_1)+\mathrm{ ht}(\alpha_2)+1}).\end{align*}$$

This follows from the observations that

$$\begin{align*}(1+V_{i\alpha_1+j\alpha_2})^{c_{ij}}\equiv 1+ c_{ij}V_{i\alpha_1+j\alpha_2} \text{ } (\mathrm{mod}\text{ } \mathrm{Fil}^{\mathrm{ht}(\alpha_1)+\mathrm{ht}(\alpha_2)+1})\end{align*}$$

by degree calculation and that $\deg (V_{i\alpha _1+j\alpha _2}) + \deg (V_{\alpha _k})> \mathrm {ht}(\alpha _1)+\mathrm {ht}(\alpha _2) $ for $k\in \{1, 2\}$ . This shows (R-5’).

Next consider the relation (R-7). For $\alpha \in \Phi ^{+}, \beta \in \Phi ^{-}$ , $\alpha +\beta \in \Phi \backslash \{0\}$ ,

$$\begin{align*}(1+V_{\alpha})(1+U_{\beta})=\prod_{\stackrel{i, j>0}{i\alpha+j\beta \in \Phi^{-}}}(1+U_{i\alpha+j\beta})^{c_{ij}p^{j-1}}\prod_{\stackrel{i, j>0}{i\alpha+j\beta \in \Phi^{+}}}(1+V_{i\alpha+j\beta})^{c_{ij}p^j}(1+U_{\beta})(1+V_{\alpha}).\end{align*}$$

Let us first deal with the case $i\alpha + j\beta \in \Phi ^{-}$ . For $ \deg U_{i\alpha +j\beta }^m=m(h+ \mathrm {ht}(i\alpha +j\beta ))> h+ \mathrm {ht}(\beta )+\mathrm {ht}(\alpha )$ , we have $ U_{i\alpha +j\beta }^m=0 \text { }(\mathrm {mod}\text { } \mathrm {Fil}^{h+ \mathrm {ht}(\beta )+\mathrm {ht}(\alpha )+1}). $ Hence we can restrict to the case $m(h+ \mathrm {ht}(i\alpha +j\beta )) \leq h+ \mathrm {ht}(\beta )+\mathrm {ht}(\alpha ).$

$$\begin{align*}\begin{array}{ccccc} m(h+ \mathrm{ht}(i\alpha+j\beta)) &\le& h+ \mathrm{ht}(\beta)+\mathrm{ht}(\alpha) \\ \text{gives}\; m &\le& \dfrac{h+ \mathrm{ht}(\beta)+\mathrm{ht}(\alpha)}{h+ \mathrm{ht}(i\alpha+j\beta)}\\ &\le& 2h -2 \;(\text{as } h+ \mathrm{ht}(i\alpha+j\beta)\ge 1, \mathrm{ht}(\beta) \leq -1, \mathrm{ht}(\alpha)\leq h-1)\\ &\le& 2(p-1)-2= 2p-4 \;(\text{as } p>1+h). \end{array}\end{align*}$$

Similar to the argument we gave while deducing (R-6’), we see that if $j-1 \ge 2$ , then with this restriction on m, none of the corresponding terms will contribute modulo the filtration as the coefficients become trivial modulo p.

When $j-1=1$ or $j=2$ then we potentially will have contribution from $m=p$ term. In that case we have, $\deg (U_{i\alpha + 2\beta })^p= p(h+ \mathrm {ht}(i\alpha +2\beta ))> (h+1)(h+ \mathrm {ht}(i\alpha +2\beta ))$ .

Note that as $h+\mathrm {ht}(i\alpha +2\beta )\geq 1$ , $(h+1)(h+ \mathrm {ht}(i\alpha +2\beta ))\geq (\textit {h}+1)$ . So, of course, $(h+1)(h+ \mathrm {ht}(i\alpha +2\beta ))> \textit {h}+ \mathrm {ht}(\alpha )+ \mathrm {ht}(\beta )$ if $\mathrm {ht}(\alpha ) +\mathrm {ht}(\beta ) < 1$ . So we assume $\mathrm {ht}(\alpha )+ \mathrm {ht}(\beta ) \ge 1$

$$\begin{align*}\begin{array}{ccccc} \mathrm{ht}(\alpha)+ \mathrm{ht}(\beta) &\ge& 1 \\ \text{hence } \text{ } \text{ } i\mathrm{ht}(\alpha) &\ge& i(1-\mathrm{ht}(\beta))\\ \text{which gives } \text{ } \text{ } i\mathrm{ht}(\alpha)+ 2\mathrm{ht}(\beta) &\ge& i(1-\mathrm{ht}(\beta))+2\mathrm{ht}(\beta)= \textit{i}+ (2-\textit{i})\mathrm{ht}(\beta). \end{array}\end{align*}$$

So if $2-i\le 0$ then the righthand side is definitely $>0$ , but this is not possible as ${i\alpha +2\beta }$ was assumed to be a negative root. Thus we see that only value i can take is $1$ . But in that case as $p>1+h$ ,

$$ \begin{align*} p(h + \mathrm{ht}(\alpha)+ 2\mathrm{ht}(\beta))&>(h+1)(h + \mathrm{ht}(\alpha)+ 2\mathrm{ht}(\beta)) \\ &=(h + \mathrm{ht}(\alpha)+\mathrm{ht}(\beta)) + h\big(h + \mathrm{ht}(\alpha)+ 2\mathrm{ht}(\beta)\big)+\mathrm{ht}(\beta)\\ &=(h + \mathrm{ht}(\alpha)+\mathrm{ht}(\beta)) + h\big(h + \mathrm{ht}(\alpha+2\beta)\big)+\mathrm{ht}(\beta)\\ &\geq (h + \mathrm{ht}(\alpha)+\mathrm{ht}(\beta))+(h+\mathrm{ht}(\beta))\text{ } (\text{as } h+\mathrm{ht}(\alpha+2\beta) \geq 1)\\ &>h + \mathrm{ht}(\alpha)+\mathrm{ht}(\beta) \text{ } (\text{as } h+\mathrm{ht}(\beta) \geq 1). \end{align*} $$

When $j=1$ then $m(h+ \mathrm {ht}(i\alpha +\beta )) \geq \textit {h}+ \mathrm {ht}(\alpha )+\mathrm {ht}(\beta )$ for all $ m$ , with equality only when $m=1, i=1$ . Thus the only contribution we can possibly have is ${(1+c_{11} U_{\alpha +\beta })}$ .

Now to look at the contribution coming from positive roots part of the equation (R-7), i.e., from the terms $\underset {\underset {i\alpha +j\beta \in \Phi ^{+}}{i, j>0}}{\prod }(1+V_{i\alpha +j\beta })^{c_{ij}p^j}.$

By the same logic as above (i.e., bounding m by $2p-4$ , then showing that the binomial coefficients become trivial modulo p) this time also $j\ge 2$ gets ruled out immediately. The calculation is similar, hence omitted.

When $j=1$ we are again concerned with the $m=p$ term only.

In this case we have

$$ \begin{align*} \deg(V_{i\alpha+ \beta})^p= p\big( \mathrm{ht}(i\alpha+\beta)\big)&>(h+1) \big(i\mathrm{ht}(\alpha)+\mathrm{ht}(\beta)\big)\\ &=i\mathrm{ht}(\alpha) +\mathrm{ht}(\beta) +h \big(\mathrm{ht}(i\alpha+\beta)\big)\\ &\geq h+\mathrm{ht}\alpha+\mathrm{ht}(\beta) \text{ } (\text{as } i>0, \mathrm{ht}(i\alpha+\beta)>0). \end{align*} $$

Thus relation (R-7) reduces to

$$\begin{align*}(1+V_\alpha)(1+ U_\beta)= (1+c_{11} U_{\alpha+\beta})(1+ U_\beta)(1+ V_\alpha) \text{ }(\mathrm{mod}\text{ } \mathrm{Fil}^{h+ \mathrm{ht}(\beta)+\mathrm{ht}(\alpha)+1}), \;\text{when}\;\alpha+\beta\in \Phi.\end{align*}$$

Now, $\deg (U_{\alpha +\beta }V_\alpha )=h+2\mathrm {ht}(\alpha )+\mathrm {ht}(\beta )>h+ \mathrm {ht}(\beta )+\mathrm {ht}(\alpha )$ and

$$ \begin{align*} \deg(U_{\alpha+\beta}U_\beta)&=2h+\mathrm{ht}(\alpha) +2\mathrm{ht}(\beta)\\ &=h+ \mathrm{ht}(\beta)+\mathrm{ht}(\alpha)+(h+\mathrm{ht}(\beta))\\ &>h+ \mathrm{ht}(\beta)+\mathrm{ht}(\alpha) \text{ }(\text{as } h+\mathrm{ht}(\beta) \geq 1). \end{align*} $$

Hence, after further simplifying we obtain

$$\begin{align*}V_\alpha U_\beta= c_{11}U_{\alpha+\beta}+U_\beta V_\alpha\text{ }(\mathrm{mod}\text{ } \mathrm{Fil}^{h+ \mathrm{ht}(\beta)+\mathrm{ht}(\alpha)+1})\end{align*}$$

which is exactly relation (R-7’).

Now consider relation (R-8). First, we will show that $(1+U_{-\alpha })^Q=(1+U_{-\alpha })\text { }(\mathrm {mod}\text { } \mathrm {Fil}^{h+1})$ . If $(h+\mathrm {ht}(-\alpha ))m>h$ then $U_{-\alpha }^m=0\text { }(\mathrm {mod}\text { } \mathrm {Fil}^{h+1}).$ So we need to worry about the case $m \geq 2$ and $(h+\mathrm {ht}(-\alpha ))m \leq h$ . But then again $m \leq h<p-1$ and so ${Q \choose m} =0\text { } (\mathrm {mod}\text { } p)$ since $Q=1\text { } (\mathrm {mod}\text { } p)$ .

Next, we will show that $(1+V_{\alpha })^Q=(1+V_{\alpha })\text { }(\mathrm {mod}\text { } \mathrm {Fil}^{h+1})$ . If $m\mathrm {ht}(\alpha )>h$ , the the degree of $V_{\alpha }^m$ is strictly larger than h. If $m\mathrm {ht}(\alpha ) \leq h$ , then $m \leq h< p-1$ and so again ${Q \choose m} =0\text { } (\mathrm {mod}\text { } p)$ . Hence relation (R-8) reduces to

$$ \begin{align*}(1+V_\alpha)(1+ U_{-\alpha})= (1+ U_{-\alpha})\underset{\delta_i\in \Delta}{\prod}(1+ W_{\delta_i})^{n_i }(1+V_\alpha)\text{ }(\mathrm{mod}\text{ } \mathrm{Fil}^{h+1}).\end{align*} $$

Since $\deg (W_{\delta _i})=h$ and $ n_i \geq 0$ , further simplification yields

$$ \begin{align*}V_\alpha U_{-\alpha} = \underset{\delta_i \in \Delta}{\sum}n_iW_{\delta_i}+U_{-\alpha}V_\alpha \text{ } (\mathrm{mod}\text{ } \mathrm{ Fil}^{h+1}).\\[-44pt]\end{align*} $$

5.3 Computing inversions modulo filtrations

Recall that $\{X_1,\cdots , X_d \}$ was an ordered set of variables generating the noncommutative p-adic power series ring $\mathcal {A}$ . We set $X^i:=X_{i_1}\cdots X_{i_w}$ to mean the monomial $X_{i(1)}\cdots X_{i(w)}$ . Then we can change $X^i$ to a well-ordered monomial ( $b \rightarrow i_b$ increasing) by a sequence of transpositions. Consider a move $(b,b+1) \rightarrow (b+1,b)$ and assume $i_b>i_{b+1}.$ We write

$$ \begin{align*}X^i=X^fX_{i_b}X_{i_{b+1}}X^e,\end{align*} $$

and we look at $X^i \text { } (\mathrm {mod}\text { } \mathrm {Fil}^{\deg (X^i)+1}).$ Suppose $(X_{i_b},X_{i_{b+1}})$ is any pair of variables among $\{U_\beta , W_{\delta _i}, V_\alpha , \beta \in \Phi ^-, \delta _i \in \Delta , \alpha \in \Phi ^+\}$ but in the wrong ordering (for example, let $(X_{i_b},X_{i_{b+1}})=(V_\alpha , W_\delta )$ from relation $\textbf {(R-2')}$ ). From Proposition 5.2.1, we see that variables in the wrong ordering commute modulo filtration of an appropriate degree except relations (R-5’), (R-7’) and (R-8’). Hence, except for these three relations,

$$ \begin{align*}X_{i_b}X_{i_{b+1}}=X_{i_{b+1}}X_{i_b}\text{ } (\mathrm{mod}\text{ } \mathrm{Fil}^{\deg(X_{i_b})+\deg(X_{i_{b+1}})+1}).\end{align*} $$

Therefore, we can reduce the number of inversions and change the wrong ordered variables into right order, modulo suitable filtration. For the three relations (R-5’), (R-7’), and (R-8’), we have to compute the number of inversions explicitly.

Lemma 5.3.1 Suppose $(X_{i_b},X_{i_{b+1}})$ be either of the three pairs $(V_{\alpha _1},V_{\alpha _2}), (V_\alpha ,U_\beta )$ or $(V_\alpha ,U_{-\alpha })$ in the wrong ordering coming from left hand side of the relations (R-5’), (R-7’), and (R-8’) satisfying equations of the form

$$ \begin{align*}X_{i_b}X_{i_{b+1}}=X_{i_{b+1}}X_{i_b}+ \text{ linear terms in } X_i \text{ } (\mathrm{mod}\text{ } \mathrm{Fil}^{\deg(X_{i_b})+\deg(X_{i_{b+1}})+1}).\end{align*} $$

Then the number of inversions in $X^fX_{i_{b+1}}X_{i_b}X^e$ and $X^fX_iX^e$ are both strictly less than the number of inversions of $X^i$ . That is, the two moves $X_{i_b}X_{i_{b+1}} \rightarrow X_{i_{b+1}}X_{i_b},$ and $X_{i_b}X_{i_{b+1}} \rightarrow X_i$ reduces the number of inversions modulo $\mathrm {Fil}^{\deg (X_{i_b})+\deg (X_{i_{b+1}})+1}$ .

Proof First let us compute the number of inversions of $X^i$ . Let $\xi $ and $\mu $ be denote indices $\neq i_b, i_{b+1}$ in the product $X^i$ .

Consider relation (R-5’). The number of inversions in $X^i$ was originally as follows:

$$ \begin{align*}\mathrm{inv}=\underset{\underset{i_\xi>i_\mu}{\xi<\mu}}{\sum}1+\underset{\underset{i_\mu \in [\mathrm{I}(V_{\alpha_2}),\mathrm{I}(V_{\alpha_1}))}{\mu>b+1}}{\sum}1+\Big(2\underset{\underset{i_\mu<\mathrm{I}(V_{\alpha_2})}{\mu>b+1}}{\sum}1\Big)+\underset{\underset{i_\xi \in (\mathrm{I}(V_{\alpha_2}),\mathrm{I}(V_{\alpha_1})]}{\xi<b}}{\sum}1+\Big(2\underset{\underset{i_\xi>\mathrm{I}(V_{\alpha_1})}{\xi<b}}{\sum}1\Big)+1,\end{align*} $$

where $\mathrm {I}(*):=\mathrm {index}(*)$ is the index of the variable $*$ .

As ${\mathrm {I}}(V_{\alpha _2}) <{\mathrm {I}}(V_{\alpha _1})$ , the transition $V_{\alpha _1}V_{\alpha _2} \rightarrow V_{\alpha _2}V_{\alpha _1}$ clearly reduces the number of inversions, and by changing $V_{\alpha _1}V_{\alpha _2}$ to $c_{ij}V_{i{\alpha _1}+j{\alpha _2}}$ we get

$$ \begin{align*}{\mathrm{inv}^{\prime}}=\underset{\underset{i_\xi>i_\mu}{\xi<\mu}}{\sum}1+\underset{\underset{i_\mu<\mathrm{I}(V_{i{\alpha_1}+j{\alpha_2}})}{\mu>b+1}}{\sum}1+\underset{\underset{i_\xi>\mathrm{I}(V_{i{\alpha_1}+j{\alpha_2}}).}{\xi<b}}{\sum}1\end{align*} $$

Because of the choice of an additive ordering on the set of positive roots in Section 2.3,

(5.3.1) $$ \begin{align} {\mathrm{I}}(V_{\alpha_2}) < {\mathrm{I}}(V_{i{\alpha_1}+j{\alpha_2}}) <{\mathrm{I}}(V_{\alpha_1}).\end{align} $$

Hence the second term of ${\mathrm {inv}^{\prime }}$ is

$$ \begin{align*} \underset{\underset{i_\mu<\mathrm{I}(V_{i{\alpha_1}+j{\alpha_2}})}{\mu>b+1}}{\sum}1 \leq \underset{\underset{i_\mu<\mathrm{I}(V_{{\alpha_1}})}{\mu>b+1}}{\sum}1=\underset{\underset{i_\mu \in [\mathrm{I}(V_{\alpha_2}),\mathrm{I}(V_{\alpha_1}))}{\mu>b+1}}{\sum}1+\underset{\underset{i_\mu<\mathrm{I}(V_{\alpha_2})}{\mu>b+1}}{\sum}1 \leq \underset{\underset{i_\mu \in [\mathrm{I}(V_{\alpha_2}),\mathrm{I}(V_{\alpha_1}))}{\mu>b+1}}{\sum}1+\Big(2\underset{\underset{i_\mu<\mathrm{I}(V_{\alpha_2})}{\mu>b+1}}{\sum}1\Big), \end{align*} $$

and the third term of ${\mathrm {inv}^{\prime }}$ is

$$ \begin{align*} \underset{\underset{i_\xi>\mathrm{I}(V_{i{\alpha_1}+j{\alpha_2}})}{\xi<b}}{\sum}1 \leq \underset{\underset{i_\xi>\mathrm{I}(V_{{\alpha_2}})}{\xi<b}}{\sum}1 = \underset{\underset{i_\xi \in (\mathrm{I}(V_{\alpha_2}),\mathrm{I}(V_{\alpha_1})]}{\xi<b}}{\sum}1+\underset{\underset{i_\xi>\mathrm{I}(V_{\alpha_1})}{\xi<b}}{\sum}1 \leq \underset{\underset{i_\xi \in (\mathrm{I}(V_{\alpha_2}),\mathrm{I}(V_{\alpha_1})]}{\xi<b}}{\sum}1+\Big(2\underset{\underset{i_\xi>\mathrm{I}(V_{\alpha_1})}{\xi<b}}{\sum}1\Big). \end{align*} $$

Hence ${\mathrm {inv}^{\prime }}<\mathrm {inv}.$ for the relation (R-5’).

The proofs for the relations (R-7’) and (R-8’) are similar and hence omitted, the key necessary things corresponding to (5.3.1) being ${\mathrm {I}}(U_\beta ) <{\mathrm {I}}(U_{\alpha +\beta }) <{\mathrm {I}}(V_\alpha )$ for the relation (R-7’) and ${\mathrm {I}}(U_{-\alpha }) <{\mathrm {I}}(W_{\delta _i}) <{\mathrm {I}}(V_\alpha )$ for the relation (R-8’).

One notices that the main idea of this proof can be summarized as follows.Footnote ¹ It is clear that $X^fX_{i_{b+1}}X_{i_b}X^e$ has exactly one less inversion than $X^i=X^fX_{i_b}X_{i_{b+1}}X^e$ . Therefore, it suffices to show that the number of inversion pairs of $X^fX_iX^e$ is no more than that of $X^fX_{i_b}X_{i_{b+1}}X^e$ . But one can directly see that if $(X,X_i)$ (resp. $(X_i,X)$ ) is an inversion pair with X from $X^f$ (resp. with X from $X^e$ ), then at least one of $(X,X_{i_b})$ and $(X,X_{i_{b+1}})$ (resp. one of $(X_{i_b},X)$ and $(X_{i_{b+1}},X)$ ) is an inversion pair due to our choice of the ordered basis.

Our main theorems below will now follow from the exact same proofs as in [Reference ClozelClo11, p. 554, Theorems 1.2 and 1.4].