Arithmetic of Bideterminants

The calculus of bideterminants is needed inside ${\cal K}=A^{{\rm sh}}_{R,q}(n,r)$. Unless otherwise stated the rules hold in $A^{{\rm s}}_{R,q}(n,r)$ too. Recall the definition of $\kappa_{\lambda}$ from section 3.

Lemma 9.1   Let $\lambda \in \Lambda(p, r)$ be a composition and $y=q^2$. Then to each $i<r$ such that the simple transposition $s_i=(i,i+1)$ is contained in the standard Young-subgroup ${\cal S}_{\lambda}$, there are endomorphisms $\mu_{\lambda,i}, \mu'_{\lambda,i}\in {\rm End}_{R}{(V^{\otimes r})}$ satisfying

$$\kappa_{\lambda}=({\rm id}_{V^{\otimes r}}-y^{-1}\beta_i)\mu_{\lambda ,i}= \mu'_{\lambda,i}({\rm id}_{V^{\otimes r}}-y^{-1}\beta_i).$$

PROOF: Let us first reduce to the case $\lambda =\alpha_r=(r)$. Setting $\kappa_r:=\kappa_{\alpha_r}$, $k_s:=\lambda_1 + \ldots + \lambda_{s-1}$, $\mu_{r,i}:=\mu_{\alpha_r,i}$, $\mu'_{r,i}:=\mu'_{\alpha_r,i}$ and

$$\kappa_{\lambda}^s:={\rm id}_{V^{\otimes k_s}} \otimes \kappa_{\lambda_s}\otimes{\rm id}_{V^{\otimes r-\lambda_s-k_s}},$$

we can extend the definition to arbitrary $\lambda$ by using the formula $\kappa_{\lambda}=\kappa_{\lambda}^1\kappa_{\lambda}^2\ldots \kappa_{\lambda}^p$ in which the factors commute. Now, using standard reduced expressions for permutations $w \in {\cal S}_{r}$ one easily verifies the following recursion rules for $r>1$:

$$\kappa_r=\kappa_{r-1}({\rm id}_{V^{\otimes r}}+ \sum_{l=1}^{r-1}(-y)^{l-r}\beta_{r-1}\beta_{r-2} \ldots \beta_l)=$$

$$({\rm id}_{V^{\otimes r}}+ \sum_{l=1}^{r-1}(-y)^{l-r}\beta_l\beta_{l+1} \ldots \beta_{r-1})\kappa_{r-1}.$$

We proceed by induction on $r$, the case $r=2$ being clear. The case $i<r-1$ can be handled immediately with the help of the above recursion formula. If $i=r-1$ we calculate

$$\kappa_r = \kappa_{r-1}({\rm id}_{V^{\otimes r}}-y^{-1}\beta_{r-1... ...eta_{r-2}) \sum_{l=1}^{r-2}(-y)^{l-r}\beta_{r-1}\beta_{r-2} \ldots \beta_l.$$

But by the braid relations we get

$$({\rm id}_{V^{\otimes r}}-y^{-1}\beta_{r-2})(-y)^{l-r}\beta_{r-1}... ...a_{r-1}\beta_{r-2}\ldots \beta_l({\rm id}_{V^{\otimes r}}-y^{-1}\beta_{r-1}),$$

yielding the right hand side factorization of $\kappa_r$. The other formula is obtained similarly. $\Box$

Corollary 9.2   Let ${\bf j} \in I(n,r)$ be a multi-index possessing two identical neighbouring indices $j_l=j_{l+1}$ and $\lambda \in \Lambda(p, r)$ such that the transposition $s_l$ is contained in ${\cal S}_{\lambda}$. Then $t_q^{\lambda}({\bf i}:{\bf j})=t_q^{\lambda}({\bf j}:{\bf i})=0$ holds for all ${\bf i} \in I(n,r)$.

PROOF: By assumption, $v_{{\bf j}}$ lies in the kernel of $({\rm id}_{V^{\otimes r}}-y^{-1}\beta_l)$. Consequently the assertion concerning $t_q^{\lambda}({\bf i}:{\bf j})$ follows immediately from Lemma 9.1 since $t_q^{\lambda}({\bf i}:{\bf j})= x_{{\bf i} {\bf j}}\wr \kappa_{\lambda}$. Using the matrix transposition map $^*$ introduced in section 8, the formula for exchanged multi-indices follows as well. $\Box$

Next, we investigate the transition from $A^{{\rm s}}_{R,q}(n)$ to its epimorphic image $A^{{\rm sh}}_{R,q}(n)$. For this purpose denote by ${{\cal G}_{r}}$ the ideal generated by $G:=\gamma_1=\gamma\otimes {\rm id}_{V^{\otimes r-2}}$ in the algebraic span ${\cal A}_r$ of the endomorphisms $\beta_i$ and $\gamma_i$. By equation (1) the realtion $\beta^2=(q^2-1)\beta + q^2{\rm id}_{V^{\otimes r}2}$ holds in ${\cal A}_r/ {\cal G}_{r}$. By the braid relations $\beta_i\beta_{i+1}\beta_i=\beta_{i+1}\beta_i\beta_{i+1}$ the relations $\beta_i^2=(q^2-1)\beta_i + q^2{\rm id}_{V^{\otimes r}2}$ and $\gamma_i = 0$ must hold in ${\cal A}_r/ {\cal G}_{r}$ for all $i$ as well. The Iwahori-Hecke algebra ${{\cal H}_q(r)}$ of type $A$ is defined on generators $T_{s_i}$ for $i \in \{1, \cdots, r-1\}$ by relations

$$\begin{array}{rll} T_{s_i}T_{s_j} & = T_{s_j}T_{s_i} & \m... ...& = (q^2-1)T_{s_1} + q^2 & \mbox{ where } i < r. \end{array}$$

Therefore, there is an epimorphism from ${\cal H}_q(r)$ to the quotient ${\cal A}_r/ {\cal G}_{r}$ sending the generator $T_{s_l}$ to $\beta_l +{\cal G}_{r}$ (notation as in [DD]).

Lemma 9.3   Let $A, B\in {\cal A}_r$ be endomorphisms of $V^{\otimes r}$ such that $A\equiv B$ modulo ${\cal G}_{r}$. Then, the equation $x_{{\bf i} {\bf j}}\wr A=x_{{\bf i} {\bf j}}\wr B$ holds in ${\cal K}$.

PROOF: We have to show that $x_{{\bf i} {\bf j}}\wr A=0$ for all $A\in {\cal G}_{r}$. Let $F,H\in {\cal A}_r$ be such that $A=FGH$. From the defining equation of the quantum coefficient of dilation $d_q$ from section 4 we have

$$x_{{\bf i} {\bf j}}\wr G= \left\{ \begin{array}{ll} 0 & j_1'\ne... ...}\ldots x_{i_r j_r} & j_1'=j_2 \mbox{ and } i_1'=i_2.\\ \end{array} \right.$$

This means $x_{{\bf i} {\bf j}}\wr G=0$ in ${\cal K}$ for all ${\bf i},{\bf j}\in I(n,r)$. By (5) we have $x_{{\bf i} {\bf j}}\wr F=F\wr x_{{\bf i} {\bf j}}$ and therefore,

$$x_{{\bf i} {\bf j}}\wr FGH=\sum_{{\bf k},{\bf l},{\bf s} \in I(n,... ...\in I(n,r)} f_{{\bf i}{\bf k}}(x_{{\bf k} {\bf s}}\wr G)h_{{\bf s}{\bf j}}= 0.$$

where $(f_{{\bf i}{\bf j}})_{{\bf i}, {\bf j} \in I(n,r)}$, $(g_{{\bf i}{\bf j}})_{{\bf i}, {\bf j} \in I(n,r)}$ and $(h_{{\bf i}{\bf j}})_{{\bf i}, {\bf j} \in I(n,r)}$ are the coefficient matrices of $F, \; G$ and $H$. $\Box$

We extend the notation introduced in (2). Let $\mu \in {\rm End}_{R}{(V^{\otimes r})}$ be an endomorphism of $V^{\otimes r}$. Set

$$\begin{array}{rl} t_q^{\lambda}({\bf i}:{\bf j})\wr \mbox{\i... ...})=(\mu\kappa_{\lambda}) \wr x_{{\bf i} {\bf j}}. \end{array}$$ (10)

Similar expressions are used with respect to the capital $T$ notation for bideterminants.

Lemma 9.4   For all ${\bf i},{\bf j}\in I(n,r)$ and $w \in {\cal S}_{\lambda}$ the following equations hold in ${\cal K}$;

$$\beta (w)\wr t_q^{\lambda}({\bf i}:{\bf j})= \beta (w)^{-1}\wr t... ...bf i}:{\bf j})\wr \beta (w)^{-1} =t_q^{\lambda}({\bf i}:{\bf j})\wr \beta (w)$$

PROOF: Modulo ${\cal G}_{r}$ we have

$$\beta (w)\kappa_{\lambda}=\beta (w)^{-1}\kappa_{\lambda}= (-1)^{... ...kappa_{\lambda} = \kappa_{\lambda} \beta (w)^{-1}=\kappa_{\lambda} \beta (w)$$

since the corresponding equations (where $\beta (w)$ is replaced by $T_w$) hold in the Iwahori-Hecke algebra ${\cal H}_q(r)$. Thus the assertion follows from Lemma 9.3. $\Box$

Let ${{\cal J}}$ denote the ideal in the tensor algebra ${\cal T}(V)= \bigoplus_{r \in {\mathbb{N}}\,_0} V^{\otimes r}$ generated by the twofold invariant tensor ${J}^*=\sum_{i=1}^n\epsilon_iq^{\rho_i}v_{i}\otimes v_{i'} \in V\otimes V$ and let ${{{\cal J}}_{r}}$$:={\cal J}\cap V^{\otimes r}$ be its $r$-th homogeneous summand.

Lemma 9.5   Let $U$ be the $R$-linear span of all elements $\gamma_l(v_{{\bf i}})$ where $1 \leq l<r$ and ${\bf i} \in I(n,r)$. Then $U={{\cal J}}_{r}$.

PROOF: Since $\gamma(v_{k}v_{k'})=-\epsilon_kq^{\rho_k}{J}^*$ and $\gamma(v_{k}v_{l}) =0$ for all $k, l \in \underline{r}$ with $k \neq l'$ by section 4 it follows that $U$ is contained in ${{\cal J}}_{r}$. The verification of the opposite inclusion can be reduced to consider elements of the form $v_{{\bf i}}{J}^*v_{{\bf j}}$ with ${\bf i} \in I(n,l-1), {\bf j} \in I(n,r-l-1)$ for some $1 \leq l<r$. But such an element can be written as $-\epsilon_kq^{-\rho_k}\gamma_{l}(v_{{\bf i}}v_{k}v_{k'}v_{{\bf j}})$ for some $k\in \underline{r}$. Thus the assertion follows. $\Box$

Lemma 9.6   Let $a_{{\bf j}} \in R$ be such that $\sum_{{\bf j}\in I(n,r)}a_{{\bf j}}v_{{\bf j}} \in {{\cal J}}_{r}$. Then, for all ${\bf i} \in I(n,r)$ and all compositions $\lambda$ of $r$ we have in ${\cal K}$

$$\sum_{{\bf j}\in I(n,r)} a_{{\bf j}}x_{{\bf i} {\bf j}}=0, \;\;$$    and $$\;\; \sum_{{\bf j}\in I(n,r)} a_{{\bf j}}t_q^{\lambda}({\bf i}:{\bf j})=0.$$

PROOF: First, note that the second equation follows from the first one by definition of bideterminants. By the above lemma we can reduce to the case $\sum_{{\bf j}\in I(n,r)}a_{{\bf j}}v_{{\bf j}} =\gamma_l(v_{{\bf k}})$ where ${\bf k} \in I(n,r), 1 \leq l < r$ and $a_{{\bf j}}=(\gamma_l)_{{\bf j}{\bf k}}$. Thus we get $\sum_{{\bf j}\in I(n,r)} a_{{\bf j}}x_{{\bf i} {\bf j}}= x_{{\bf i} {\bf k}}\wr \gamma_l=0.$ $\Box$

Next, we give a quantum symplectic version of Laplace duality. The corresponding classical result can be found in [Ma, 2.5.1], for instance.

Proposition 9.7 (Laplace Duality)   Let $\lambda,\mu \in\Lambda(p, r)$ be compositions, $Y$ a set of left coset representatives of ${\cal S}_{\lambda}\cap {\cal S}_{\mu}$ in ${\cal S}_{\lambda}$ and $X$ a set of right coset representatives of ${\cal S}_{\lambda}\cap {\cal S}_{\mu}$ in ${\cal S}_{\mu}$, such that $l(vw)=l(v)+l(w)$ and $l(wu)=l(u)+l(w)$ holds for all $v \in Y, u \in X$ and $w \in {\cal S}_{\lambda}\cap {\cal S}_{\mu}$. Then for all ${\bf i},{\bf j}\in I(n,r)$ the following equation holds:

$$\sum_{u \in X}(-y)^{-l(u)}t_q^{\lambda}({\bf i}:{\bf j})\wr \beta (u)= \sum_{v \in Y}(-y)^{-l(v)}\beta (v)\wr t_q^{\mu}({\bf i}:{\bf j}).$$

PROOF: Using the disjoint union

$$Z:={\cal S}_{\lambda}{\cal S}_{\mu} = \bigcup_{u\in X}{\cal S}_{\lambda}u=\bigcup_{v\in Y}v{\cal S}_{\mu} ,$$

and the fact $\beta(w)\beta(u)=\beta(wu)$, $\beta(v)\beta(w)=\beta(vw)$ which holds by length additivity we calculate

$$\begin{array}{rl} \sum_{u \in X}(-y)^{-l(u)}t_q^{\lambda}({\bf i... ...um_{v\in Y} (-y)^{-l(v)}\beta (v)\wr t_q^{\mu}({\bf i}:{\bf j}). \end{array}$$

$\Box$

The next result is needed for the transition from $t$-bideterminants of compositions to $T$-bideterminants of partitions.

Lemma 9.8   Let $\lambda \in \Lambda(p, r)$ be a composition and ${\bf i},{\bf j}\in I(n,r)$. Then the bideterminant $t_q^{\lambda}({\bf i}:{\bf j})$ can be written as a linear combination of bideterminants $T^{\lambda '}_q({\bf k}:{\bf l})$.

PROOF: First, there is a permutation $\pi \in {\cal S}_{p}$, such that $\bar{\lambda}=(\lambda_{\pi(1)}, \ldots , \lambda_{\pi(p)})\in \Lambda^+(p, r)$ is a partition. This $\bar{\lambda}$ is uniquely determined by $\lambda$ (but $\pi$ only under the restriction to be of minimal length). Clearly the parabolic subgroups ${\cal S}_{\lambda}$ and ${\cal S}_{\bar{\lambda}}$ in ${\cal S}_{r}$ are conjugate to each other. Thus, there is an element $v \in {\cal S}_{r}$ such that $v{\cal S}_{\lambda}={\cal S}_{\bar{\lambda}}v$. Furthermore, it is known from the theory of parabolic subgroups that in the left coset $v{\cal S}_{\lambda}$ and in the right coset ${\cal S}_{\bar{\lambda}}v$ there are unique representatives $w$ (resp. $\bar w$) of minimal length. Since we have $l(wu)=l(w)+l(u)$ for all $u \in {\cal S}_{\lambda}$ and $l(u\bar w)=l(u)+ l(\bar w)$ for all $u \in {\cal S}_{\bar{\lambda}}$, we have $\beta(w)\kappa_{\lambda}=\kappa_{\bar{\lambda}}\beta(\bar w)$. By the definition of bideterminants ( $t_q^{\lambda}({\bf i}:{\bf j}):=\kappa_{\lambda}x_{{\bf i} {\bf j}}$), the relations (5) holding inside $A^{{\rm s}}_{R,q}(n,r)$ and the calculus for the symbol $\wr$ given in (3) we obtain

$$\beta(w)\wr t_q^{\lambda}({\bf i}:{\bf j})= t_q^{\bar{\lambda}}(... ...j})\wr \beta(\bar w)= T^{\bar{\lambda}'}_q({\bf i}:{\bf j})\wr \beta(\bar w).$$

Since $\bar{\lambda}'=\lambda '$ this results in

$$t_q^{\lambda}({\bf i}:{\bf j})= \beta(w)^{-1}\wr T^{\lambda '}_q... ... i}{\bf k}} T^{\lambda '}_q({\bf k}:{\bf l}) \beta (\bar w)_{{\bf l}{\bf j}}.$$

$\Box$

Next, we introduce a calculus for our bideterminants bringing our special order $\lhd$ on $I(n,r)$ into the picture. First, some new notation has to be explained. The sum of two multi-indices ${\bf i} \in I(n,r)$ and ${\bf j}\in I(n,s)$ is defined by juxtaposition, that is

$${{\bf i}+{\bf j}}$$$$\mbox{\index{${{\bf i}+{\bf j}}$}}$$$$:=(i_1,\ldots , i_r, j_1, \ldots , j_s) \;\in \;I(n,r+s).$$

Note that the map $f:I(n,r)\rightarrow {\mathbb{N}}\,_0^m$ occurring in Theorem 8.3 is additive in the sense $f({\bf i}+{\bf j})= f({\bf i})+f({\bf j})$. This implies

$$f({\bf i} +{\bf j})< f({\bf i}+{\bf k})\;\; \;\; f({\bf j} +{\bf i})<f({\bf k}+{\bf i})\;\; \;\; f({\bf j})<f({\bf k})$$ (11)

with respect to the lexicographic order $<$ on ${\mathbb{N}}\,_0^m$. To a multi-index ${\bf i} \in I(n,r)$ we consider the following $R$-spans in $V^{\otimes r}$:

$${W_{{\bf i}}}$$$$\mbox{\index{${W_{{\bf i}}}$}}$$$$:= \left< v_{{\bf j}}\vert\;{\bf j} \in I(n,r) , \; f({\bf j})< f({\bf i})\right> \; \;$$ and $$\;\; {\overline W_{{\bf i}}}$$$$\mbox{\index{${\overline W_{{\bf i}}}$}}$$$$:= \left< v_{{\bf j}}\vert\;{\bf j} \in I(n,r) , \; f({\bf j}) \leq f({\bf i})\right>.$$

Furthermore, we set

$$h_{ij}:= \left\{ \begin{array}{ll} q^{-1} & \mbox{ if } j\neq i,... ...ox{ where } i'>m, \\ q^{-2} & \mbox{ if } j=i'\leq m, \end{array} \right.$$

and denote the simple transpositions by $s_l=(l,l+1)$, as before. The following lemma is the key concerning calculations with bideterminants. Again we set $y=q^2$.

Lemma 9.9   For all ${\bf i} \in I(n,r)$ and $l \in \underline{r}$ the following formulas hold in $V^{\otimes r}$ modulo the $R$-module $W_{{\bf i}}$:

$$\beta_l(v_{{\bf i}})\equiv \left\{ \begin{array}{ll} yh_{i_{l+1}... ...1}i_{l}}v_{{\bf i}s_l} & \mbox{\rm if } i_l \leq i_{l+1}, \end{array} \right.$$

$$\beta_l^{-1}(v_{{\bf i}})\equiv \left\{ \begin{array}{ll} h_{i_{... ...{l+1}i_{l}}v_{{\bf i}s_l} & \mbox{\rm if } i_l > i_{l+1}. \end{array} \right.$$

PROOF: The congruence relation for $\beta_l^{-1}$ follows from the one for $\beta_l$ because $y\beta^{-1}=\beta+(y-1)(\gamma-{\rm id}_{V^{\otimes 2}})$ by (1). Therefore, it is enough to prove the first assertion.

First, consider the case $i_l > i_{l+1}$. If $i_l\neq i_{l+1}'$, the asserted congruence relation is also an equation, as can be seen directly from the definition of $\beta$. Turning to the case $i_l=i_{l+1}'=:j\leq m$, we split ${\bf i}$ into three summands

$${\bf i}^1=(i_1, \ldots , i_{l-1}), \; \; {\bf i}^2=(j',j), \; \; {\bf i}^3=(i_{l+1}, \ldots , i_r) .$$

To $k \in \underline{n}$ we set ${\bf i}(k):={\bf i}^1 + (k,k') + {\bf i}^3$ and calculate

$$\beta_l(v_{{\bf i}}) = v_{{\bf i}s_l} + (y-1)v_{{\bf i}} -(y-1)\sum_{k>j}q^{\rho_k-\rho_j}\epsilon_k v_{{\bf i}(k)} .$$

Since $(y-1)\sum_{k=1}^nq^{\rho_k-\rho_j}\epsilon_k\epsilon_j v_{{\bf i}(k)} =(y-1)\gamma_l(v_{{\bf i}})$ we obtain the equation

$$\beta_l(v_{{\bf i}}) = v_{{\bf i}s_l} + (y-1)({\rm id}_{V^{\oti... ...\gamma_l)(v_{{\bf i}}) +(y-1)\sum_{k\leq j}q^{\rho_k-\rho_j} v_{{\bf i}(k)} .$$

But ${\bf i}(j)={\bf i}s_l$ and

$$f({\bf i}(k))=f({\bf i}^1) + f((k,k')) + f({\bf i}^3) < f({\bf i}^1) + f((j',j)) + f({\bf i}^3) =f({\bf i}(j'))=f({\bf i})$$

for all $k < j$ by (11), yielding the asserted congruence modulo $W_{{\bf i}}$. If $i_l<i_{l+1}$ the interesting case is $i_{l+1}'=i_l=:j\leq m$. Here the assertion immediately follows from the calculation

$$\beta_l(v_{{\bf i}}) = v_{{\bf i}s_l} -(y-1)\sum_{k>j'}q^{\rho_k+\rho_j} v_{{\bf i}(k)} ,$$

because $f({\bf i}(k))<f({\bf i})$ for all $k>j'$. $\Box$

Remark 9.10   By Lemma 9.5 the above lemma implies that $\overline W_{{\bf i}}+{{\cal J}}_{r}$ is invariant under ${\cal A}_r$. But, $W_{{\bf i}}=\sum_{f({\bf j})<f({\bf i})}\overline W_{{\bf j}}$ and thus, $W_{{\bf i}}+{{\cal J}}_{r}$ must be invariant as well.

Corollary 9.11   Let ${\bf j} \in I(n,r)$ and $l \in \underline{r}$. Then to each ${\bf k}\in I(n,r)$ satisfying $f({\bf k}) < f({\bf j})$ there is $a_{{\bf j}{\bf k}}(s_l)$ in $R$ (possibly zero) such that the following equations hold in ${\cal K}$ for all ${\bf i} \in I(n,r)$:

$$T^{\lambda}_q({\bf i}:{\bf j})\wr \beta_l^{-1}= h_{j_{l+1}j_l}T^... ...f k})<f({\bf j})} a_{{\bf j}{\bf k}}(s_l)T^{\lambda}_q({\bf i}:{\bf k}) \;\;$$    if $$j_l > j_{l+1},$$

$$T^{\lambda}_q({\bf i}:{\bf j})\wr \beta_l= yh_{j_{l+1}j_l}T^{\la... ...f k})<f({\bf j})} a_{{\bf j}{\bf k}}(s_l)T^{\lambda}_q({\bf i}:{\bf k}) \;\;$$    if $$j_l \leq j_{l+1}.$$

PROOF: Note that $(\beta_l^{-1})_{{\bf k} {\bf j}}$ is the coefficient of $v_{{\bf k}}$ in the expression $\beta_l^{-1}(v_{{\bf j}})$. By definition of bideterminants and the conventions (10) about $\wr$, the result follows immediately from the lemma. $\Box$

Corollary 9.12   Let ${\bf j} \in I(n,r)$ and $w\in {\cal S}_{\lambda'}$. Then there is an invertible element $a_{{\bf j}}(w) \in R$ and to each ${\bf k}\in I(n,r)$ satisfying $f({\bf k}) < f({\bf j})$ another element $a_{{\bf j}{\bf k}}(w)$ in $R$ such that the following equations hold in ${\cal K}$ for all ${\bf i} \in I(n,r)$:

$$T^{\lambda}_q({\bf i}:{\bf j})= a_{{\bf j}}(w)T^{\lambda}_q({\bf... ...f({\bf k})<f({\bf j})} a_{{\bf j}{\bf k}}(w)T^{\lambda}_q({\bf i}:{\bf k}).$$

PROOF: We use induction on the length of $w$. If this is zero there is nothing to prove. If not, we write $w=w's_l$ where $w', s_l \in {\cal S}_{\lambda'}$ and $l(w')=l(w)-1$. By the induction hypothesis we have

$$T^{\lambda}_q({\bf i}:{\bf j})= a_{{\bf j}}(w')T^{\lambda}_q({\b... ...({\bf k})<f({\bf j})} a_{{\bf j}{\bf k}}(w') T^{\lambda}_q({\bf i}:{\bf k}).$$

But by Lemma 9.4 we have $T^{\lambda}_q({\bf i}:{\bf j}w')= -T^{\lambda}_q({\bf i}:{\bf j}w')\wr \beta_l^{-1}$ as well as $T^{\lambda}_q({\bf i}:{\bf j}w')= -T^{\lambda}_q({\bf i}:{\bf j}w')\wr \beta_l$. Thus, the assertion follows from the preceding corollary and the fact that $f({\bf j})=f({\bf j}w')$. $\Box$

Lemma 9.13   Let ${\bf i} \in I(n,r)$ satisfy $i_1\leq i_2 \leq \ldots \leq i_r$ and let $w \in {\cal S}_{r}$ be arbitrary. Then the following congruence relation holds in $V^{\otimes r}$ modulo the $R$-submodule $W'_{{\bf i}}=W_{{\bf i}}+{{\cal J}}_{r}$:

$$\beta(w^{-1})(v_{{\bf i}})\equiv y^{l(w)}h_{{\bf i}}(w)v_{{\bf i}w} .$$

Here we have set $h_{{\bf i}}(w):=\prod h_{i_{w(k)}i_{w(j)}}$, where the product runs over all pairs $1\leq j < k \leq r$ such that $w(j)>w(k)$.

PROOF: We use induction on $r$, the case $r=1$ being trivial. For $r>1$ we embed ${\cal S}_{r-1}$ as the parabolic subgroup of ${\cal S}_{r}$ generated by $s_1, \ldots ,s_{r-2}$, which fix $r$. If $w\in {\cal S}_{r-1}$ there is nothing to prove by the induction hypothesis. Otherwise, we write $w=w's$ where $w'\in {\cal S}_{r-1}$ and $s:=s_{r-1} s_{r-2}\ldots s_{j+1}s_j$ for an appropriate $j < r$, thus $l(w)=l(w')+r-j$. By the induction hypothesis, Lemma 9.9 and Remark 9.10 we calculate

$$\begin{array}{rl} \beta(w^{-1})(v_{{\bf i}}) & \equiv \beta(s^{... ...)}} v_{{\bf i}w's} \\ & = y^{l(w)}h_{{\bf i}}(w)v_{{\bf i}w}. \end{array}$$

$\Box$

Corollary 9.14   Let ${\bf j} \in I(n,r)$ satisfy $j_l \leq j_{l+1} \leq \ldots \leq j_{k-1} \leq j_k$ for some $1\leq l < k \leq r$ and $w \in {\cal S}_{r}$ satisfy $w(i)=i$ for $1\leq i \leq l$ or $k< i \leq r$. Then, to each ${\bf k}\in I(n,r)$ satisfying $f({\bf k}) < f({\bf j})$ there is an element $a'_{{\bf j}{\bf k}}(w)$ in $R$ such that the following equations hold in ${\cal K}$ for all ${\bf i} \in I(n,r)$:

$$T^{\lambda}_q({\bf i}:{\bf j})\wr \beta(w)= y^{l(w)}h_{{\bf j}}(... ...{f({\bf k})<f({\bf j})} a'_{{\bf j}{\bf k}}(w)T^{\lambda}_q({\bf i}:{\bf k}).$$

PROOF: As for the proof of Corollary 9.11, this follows easily from the preceding lemma, Lemma and the definition of bideterminants. $\Box$