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\begin{document}
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\begin{slide}
\null
\vspace*{2cm}
\begin{flushleft}
{\color{zvyr}\rule{\textwidth}{1cm}}

\smallskip

{\Huge\bfseries\sffamily\color{head} \MakeUppercase{Youngovy
diagramy, tabulky a jejich u�it�}}

\smallskip

{\color{zvyr}\rule{\textwidth}{1cm}}
\end{flushleft}
\end{slide}
\begin{slide}
\begin{thebibliography}{Kraft-Procesi}
\bibitem[Mathworld]{ew} 
Eric W. Weisstein. "Partition Function $P$.", \\MathWorld -- A~Wolfram Web Resource. 
\texttt{http://mathworld.wolfram.com/PartitionFunctionP.html}
\bibitem[Sternberg]{st} Shlomo Sternberg, \textit{Group theory and physics},
Cambridge University Press, 1994, ISBN~0~521~55885~9
\bibitem[Wybourne]{wy} Brian G. Wybourne, Symmetric functions and the symmetric group,
Lecture Notes, \texttt{http://www.phys.uni.torun.pl/\~{}bgw/Symfn.html}
\bibitem[Fulton]{fu} William Fulton, \textit{Young tableaux}, Cambridge
University Press, 1997, ISBN~0~521~56724~6
\bibitem[King]{ki} Ronald C. King, Generalized Young tableaux and the general
linear group, Journal of Mathematical Physics (11), leden 1970
\bibitem[Kraft-Procesi]{kp} Hanspeter Kraft, Claudio Procesi, \textit{Invariant theory -- 
A~primer}, preprint, 1996.
\end{thebibliography}
\end{slide}
\begin{slide}
\section{Motivace}
Jist� konkr�tn� probl�m ve varia�n�m po�tu vy�aduje rozklad soubor�
veli�in indexovan�ch horn�mi a doln�mi indexy na bezestop� ��sti.
\pause

Diferenci�ln� $q$-forma na $r$-t�m prodlou�en� fibrovan� variety
$Y\zob X$ je zaps�na v~adaptovan�ch sou�adnic�ch takto
$$
\rho = \sum_{p=0}^q A{}^{J_1}_{\sigma_1} \cdots {}^{J_k}_{\sigma_k}{\;}_{j_{p+1}\cdots j_q} \dd
y^{\sigma_1}_{J_1} \ww \ldots \ww \dd y^{\sigma_p}_{J_p} \ww \dd
x^{j_{p+1}} \ww \ldots \ww \dd x^{j_q}.
$$
Z~hlediska varia�n�ho po�tu jsou podstatn� pouze bezestop� komponenty.
Komponenty obsahuj�c� stopu jsou varia�n� trivi�ln�.
\pause

Takov� lok�ln� rozklad lze prov�st pon�kud obecn�ji na $r$-jetov�ch
prodlou�en�ch $n$-rozm�rn�ch podvariet
$(n+m)$-rozm�rn� variety $Y$.
\pause

P�esto�e veli�iny v~p�vodn�m probl�mu nemaj� tenzorov� charakter,
vykazuje probl�m \emph{n�kter�} shodn� rysy s~probl�mem rozkladu tenzor� podle stop.
\end{slide}
\begin{slide}
\section{Prostor sm��en�ch tenzor�}
Uva�ujeme vektorov� prostor $V$, $\dim V=n$, du�ln� vektorov�
prostor $V^*$ a k~nim asociovan� tenzorov� prostor
$$
T^r_s V = \bigotimes^r V \otimes \bigotimes^s V^*.
$$
Tento prostor naz�v�me \emph{prostorem sm��en�ch tenzor� typu
$\binom{r}{s}$}.\pause

V~konkr�tn� b�zi $\{e_1,\ldots,e_n\}$ vektorov�ho prostoru $V$ p��eme
$$
t = t^{i_1 \cdots i_r}_{j_1\cdots j_s} e_{i_1}\otimes\cdots\otimes e_{i_r}
\otimes e^{j_1}\otimes\cdots\otimes e^{j_s}
$$
nebo ekvivalentn�
$$
t^{i_1 \cdots i_r}_{j_1\cdots j_s} = t(e^{i_1},\ldots, e^{i_r}, 
e_{j_1},\ldots, e_{j_s}).
$$
\end{slide}
\begin{slide}
\section{Stopa tenzoru}
Uva�ujme line�rn� zobrazen�
$$
C^k_\ell\colon T^r_s V \ni t \zob C^k_l t \in T^{r-1}_{s-1} V,
$$
kter� je pro libovolnou b�zi $\{e_1,\ldots,e_n\}$ vektorov�ho prostoru
$V$ a pevn� zvolen� $k\in\{1,\ldots,r\}$ a
$\ell\in\{1,\ldots,s\}$ dan� jako
$$
T^r_sV \ni t \zob C^k_\ell t = \sum_{i=1}^n
t(\ldots,\stackrel{\stackrel{\displaystyle
k}{\downarrow}}{e_i},\ldots;\ldots,\stackrel{\stackrel{\displaystyle \ell}{\downarrow}}{e^i},
\ldots)\in T^{r-1}_{s-1} V\pause
$$
Zobrazen� $C^k_l$ se naz�v� \emph{$\binom{k}{\ell}$-kontrakc�} a tenzor $C^k_l t$ se naz�v� 
\emph{ $\binom{k}{\ell}$-stopou tenzoru $t$}.

Tenzoru, jeho� v�echny kontrakce
jsou nulov�, ��k�me \emph{bezestop�}.\pause (Obecn� existuje $r.s$ r�zn�ch
kontrakc�.)
\end{slide}
\begin{slide}
\section{Rozklad tenzor� podle stop}
�loha tedy zn�:\pause
\begin{quote}
\emph{Rozlo�it obecn� tenzor $t\in T^r_s V$ na ��st
obsahuj�c� stopu a ��st bezestopou.}
\end{quote}\pause
Form�ln� m��eme ps�t
$$
t = \tilde{t} + \sum_{k=1}^r\sum_{\ell=1}^s\ide^\ell_k C^k_\ell t,
$$
kde $\tilde{t}$ je bezestop� ��st $t$ a $\ide^\ell_k$ je Kronecker�v
tenzor (sou�adnicov� vyj�d�en� je d�no Kroneckerov�m $\delta$).\pause

Takov� rozklad z�ejm� existuje. Ponechme nyn� stranou ot�zku
jeho jednozna�nosti.\pause 

Tuto proceduru lze pou��t na
rozklad podle stop ka�d�ho z~$rs$ tenzor� $C^k_\ell t\in
T^{r-1}_{s-1}V$. Postupujeme-li takto $\min(r,s)$-kr�t, na prav�
stran� z�stanou pouze sou�iny Kroneckerov�ch tenzor� a bu�
�ist� kontravariantn�ch nebo kovariantn�ch tenzor� ��du $|r-s|$; podle
toho, zda $r\le s$ nebo $r>s$.     
\end{slide}
\begin{slide}
\section{Symetrick� grupa}
Budeme nyn� studovat reprezentace symetrick� grupy ve vektorov�m
prostoru $T^k V$ �ist� kontravariantn�ch (analogicky kovariantn�ch)
tenzor�.\pause

Prvek $s\in\SM_k$ symetrick� grupy permutac� $k$ symbol� m��eme zapsat n�kolika
zp�soby. V~dal��m budeme symboly ozna�ovat jako $\{1,\ldots,
k\}$.\pause
\begin{itemize}
\item[---] \emph{p��m�} z�pis, nap�. $s\in\SM_5$
$$
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
4 & 5 & 3 & 2 & 1
\end{pmatrix}
$$\pause
\item[---] z�pis \emph{pomoc� cykl�}, nap�. pro $s$
$$
\begin{array}{lr}
1 \zob 4 \zob 2 \zob 5 \zob 1  \\
3 \zob 3
\end{array}\pause
\Longrightarrow (1425)(3)
$$\pause
\end{itemize}
\emph{Konjuga�n� t��dy} $r\sim s\Leftrightarrow \exists t\in
\SM_k\colon r=tst^{-1}$ 
jsou ur�eny permutacemi se stejnou strukturou cykl�.
\end{slide}
\begin{slide}
\section{Youngovy diagramy}
T��dy permutac� jsou tedy jednozna�n� ur�eny po�tem cykl� d�lky $1$ a�
$k$: $\#1,\ldots, \#k$. Ty jsou z�ejm� omezeny podm�nkou
$$
\#1 + 2\#2 + \cdots + k\#k = k.\pause
$$
S~v�hodou m��eme zav�st soubor ��sel $\lambda=\lambda_1,\ldots,\lambda_k$
$$
\lambda_i = \#i + \cdots + \#k, \quad \text{pro } 1 \le i \le k.
$$
Tomuto souboru ��sel $\lambda$ se ��k� rozklad $\lambda\prt k$.\pause
Ke ka�d�mu rozkladu m��eme p�i�adit Young�v diagram $[\lambda]$, nap�. pro $\SM_5$

\begin{tabular}{ccccccc}
\yng(5) & \yng(4,1) & \yng(3,2) & \yng(3,1,1)
& \yng(2,2,1) & \yng(2,1,1,1) & \yng(1,1,1,1,1) \\
$[5]$ &  $[4,1]$ & $[3,2]$ & $[3,1^2]$ & $[2^2,1]$ & $[2,1^3]$ &
$[1^5]$
\end{tabular}
\end{slide}
\begin{slide}
\section{Generov�n� rozklad�}
Program na vygenerov�n� v�ech rozklad� dan�ho ��sla:
\begin{verbatim}
from __future__ import generators
def rozklady(k):
        if k == 0:
                yield []
                return

        for p in rozklady(k-1):
                yield p + [1]
		if p and (len(p) == 1 or p[-2] > p[-1]):
                        yield p[:-1] + [p[-1] + 1]

for q in rozklady(5):
        print q,
\end{verbatim}
Jeho v�stup pro $k=5$:
\begin{verbatim}
[1, 1, 1, 1, 1] [2, 1, 1, 1] [2, 2, 1] [3, 1, 1] [3, 2] [4, 1] [5]
\end{verbatim}
\end{slide}
\begin{slide}
\section{Po�et rozklad�}
\begin{tabular}{r@{$\qquad$}|@{$\qquad$}r@{$\qquad\qquad$}r@{$\qquad$}|@{$\qquad$}r}
$k$ & Po�et rozklad� $p(k)$ & $k$ & Po�et rozklad� $p(k)$ \\
\hline
1 & 1 & 6 & 11 \\
2 & 2 & 7 & 15 \\
3 & 3 & 8 & 22 \\
4 & 5 & 20 & 627 \\
5 & 7 & 100 & 190569292
\end{tabular}\pause

\bigskip

Vytvo�uj�c� funkce pro po�et rozklad� je
$$
P(x) = \prod_{m=1}^\infty \frac{1}{1-x^m} = \sum_{k=0}^\infty p(k)
x^k.\pause
$$
Asymptotick� formule (Ramanujan, Hardy, Rademacher):
$$
p(k) \sim \frac{1}{4k\sqrt{3}} \ee^{\pi\sqrt{2k/3}}
$$
\end{slide}
\begin{slide}
\section{Reprezentace symetrick� a obecn� line�rn� grupy v~prostoru
KV tenzor�}
Pro ka�dou kone�nou grupu je po�et ireducibiln�ch reprezentac� roven 
po�tu konjuga�n�ch t��d, tj. po�tu Youngov�ch diagram� $[\lambda]$ s~$k$ �tvere�ky
($|\lambda|=k$).\pause 

\emph{Symetrick� grupa $\SM_k$} p�sob� na homogenn� prvky
$$
\SM_k\times T^k V  \ni (s,e_1 \otimes \cdots\otimes e_k) \zob
(e_{s^{-1}(1)}\otimes \cdots \otimes e_{s^{-1}(k)}) \in T^k V.\pause
$$
\emph{Obecn� line�rn� grupa $\GL(n)$} p�sob� na homogenn� prvky
$$
\GL(n) \times T^k V \ni (A,e_1 \otimes \cdots\otimes e_k) \zob
(Ae_1\otimes \cdots \otimes Ae_k) \in T^k V.\pause
$$
Tyto dv� akce spolu z�ejm� komutuj�. Lze uk�zat, �e mezi
ireducibiln�mi reprezentacemi t�chto grup na $T^k V$ je kr�sn� vztah (Issai
Schur, 1902).\pause 
\begin{quote}
\emph{Ireducibiln� reprezentace $\SM_k$ resp. $\GL(n)$ na $T^k V$ jsou
parametrizov�ny Youngov�mi diagramy $\lambda$, $|\lambda|=k$ 
s~n�sobnostmi rovn�mi dimenzi ireducibiln� reprezentace $\GL(n)$ resp.
$\SM_k$ p��slu�n� t�mu� Youngovu diagramu.}
\end{quote}
\end{slide}
\begin{slide}
\section{Dimenze ireducibiln�ch reprezentac�}
Form�ln� tedy plat�
$$
T^k V = \sum_{|\lambda|=k} [\lambda] \otimes \{\lambda\},
$$
kde $[\lambda]$ resp. $\{\lambda\}$ jsou ireducibiln� reprezentace
$\GL(n)$ resp. $\SM_k$.\pause

Pro dimenzi $\{\lambda\}$ dostaneme
$
\displaystyle
\dim\{\lambda\} = \frac{k!}{\Hook(\lambda)},
$\par
kde $\Hook(\lambda)$ je sou�in d�lek v�ech h��k�.

Pro dimenzi $[\lambda]$ z�sk�me
$
\displaystyle
\dim [\lambda] = \frac{\prod_{i,j\in\lambda}(n+j-i)}{\Hook(\lambda)}
$.\pause
$$
\Hook\left({\tiny\yng(4,3,1)}\right) = \young(6431,421,1)=576,\pause \quad 
\begin{array}{l} 
\dim \left\{{\tiny\yng(4,3,1)}\right\} = 70, \\
\dim \left[{\tiny\yng(4,3,1)}\right] = 
\frac{\text{\Huge$\prod\atop\null$}
\begin{mfpic}[15]{0}{4}{0}{3}
\tlabel[cc](0.5,2.5){$\scriptstyle n$}
\tlabel[cc](1.5,2.5){$\scriptstyle n\!+\!1$}
\tlabel[cc](2.5,2.5){$\scriptstyle n\!+\!2$}
\tlabel[cc](3.5,2.5){$\scriptstyle n\!+\!3$}
\tlabel[cc](0.5,1.5){$\scriptstyle n\!-\!1$}
\tlabel[cc](1.5,1.5){$\scriptstyle n$}
\tlabel[cc](2.5,1.5){$\scriptstyle n\!+\!1$}
\tlabel[cc](0.5,0.5){$\scriptstyle n\!-\!2$}
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\rect{(2,2),(3,3)}
\rect{(3,2),(4,3)}
\end{mfpic}
}{576}.
\end{array}
$$
\end{slide}
\begin{slide}
\section{Rozklady~ko-~a~kontravariantn�ch~tenzor�}
Rozklad $T^5 V$ na ireducibiln� komponenty vzhledem k~akci $\SM_5$.\newline 
Existuje $7$ r�zn�ch Youngov�ch diagram�
$$
\begin{tabular}{ccccccc}
\young(54321) & \young(5321,1) & \young(431,21) 
&\young(521,2,1) & \young(42,31,1) & \young(51,3,2,1) & \young(5,4,3,2,1) \\
$\dim =1$ & $\dim =4$ & $\dim =5$ & $\dim =6$ & $\dim =5$ & $\dim =4$ &
$\dim =1$
\end{tabular}\pause
$$
Celkem tedy dost�v�me
$$
T^5 V = S^5 V +
4\;{\tiny\yng(4,1)}\;V+5\;{\tiny\yng(3,2)}\;V+6_3\;{\tiny\yng(3,1,1)}\;V+
5_3\;{\tiny\yng(2,2,1)}\;V+4_4\;{\tiny\yng(2,1,1,1)}\;V+1_5\Lambda^5 V,\pause
$$
kde Young�v diagram p�sob� na tenzory jako
$$
\lambda \zob \sum_q \sum_p \sgn(q) qp, \quad 
\begin{array}{l}
\text{$p$ permutuj� ��dky diagramu,} \\ 
\text{$q$ permutuj� sloupce diagramu.}
\end{array}
$$
\end{slide}
\begin{slide}
\section{Reprezentace na~prostoru sm��en�ch tenzor�}
Budeme uva�ovat reprezentaci grupy $\GL(n)$ na prostoru
sm��en�ch tenzor� typu$\binom{r}{s}$ $T^r_s V$, tzn.
\begin{multline*}
\GL(n)\times T^r_s V~\ni (A,e_1\otimes\cdots\otimes e_r\otimes
e^1\otimes\cdots\otimes e^s) \zob\\\zob Ae_1\otimes\cdots\otimes Ae_r\otimes
A^{-1}e^1\otimes\cdots\otimes A^{-1}e^s\in T^r_s V.\pause
\end{multline*}

D�le uva�ujme reprezentaci grupy $\SM_r\times \SM_s$ na prostoru
sm��en�ch tenzor�
\begin{multline*}
\SM_r\times \SM_s\times T^r_s V~\ni (a,b,e_1\otimes\cdots\otimes e_r\otimes
e^1\otimes\cdots\otimes e^s) \zob\\\zob
e_{a^{-1}(1)}\otimes\cdots\otimes e_{a^{-1}(r)}\otimes
e^{b^{-1}(1)}\otimes\cdots\otimes e^{b^{-1}(s)}\in T^r_s V.\pause
\end{multline*}

\emph{Ukazuje se, �e ireducibiln� reprezentace $\GL(n)$ jsou reprezentace,
u~kter�ch maj� jak horn� indexy tak doln� indexy symetrie dan� Youngov�mi
diagramy $\lambda\prt r$ a $\mu\prt s$ a nav�c jsou bezestop�.}
\end{slide}
\begin{slide}
\section{Rozklad reprezentac� $\GL(n)$ v~$T^r_s V$ na ireducibiln� komponenty}
Ozna�me $[\lambda]^r$ ireducibiln� reprezentaci $\GL(n)$
na prostoru kontravariantn�ch tenzor� $T^r V$ a $[\mu]_s$
ireducibiln� reprezentaci $\GL(n)$ na prostoru kovariantn�ch tenzor� $T^s V^*$
($\lambda\prt r$ a $\mu\prt s$).\pause

Tyto reprezentace ur�uj� reprezentaci reprezentaci $\GL(n)$ na $T^r_s V$, kter�
ov�em obecn� nen� ireducibiln�. V~\cite{ki} je uveden a dok�z�n postup, kter�m
pat�i�nou ireducibiln� reprezentaci $[\lambda;\mu]^r_s$ z�sk�me. Tento postup
vede na rozklad podle stop.\pause

K~jeho odvozen� je t�eba vyu��t sdru�enosti reprezentac� na $T^r V$ a $T^s V^*$
dan� volbou jednotkov�ho objemu. V~sou�adnicov�m vyj�d�en� se tato skute�nost
projevuje jako pou��v�n� �pln� antisymetrick�ho Kroneckerova tenzoru ke
\uv{zved�n�} a \uv{sni�ov�n�} index�.
\end{slide}
\begin{slide}
\section{Sdru�enost reprezentac�}
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\gfill[white]\tlabelrect[cc](-1,6.5){$p$}
\arrow\reverse\arrow\lines{(0,9),(4,9)}
\gfill[white]\tlabelrect[cc](2,9){$q$}
\arrow\reverse\arrow\lines{(10,8),(10,3)}
\gfill[white]\tlabelrect[cc](10,5.5){$n$}
\arrow\reverse\arrow\lines{(5,9),(9,9)}
\gfill[white]\tlabelrect[cc](7,9){$q$}
\arrow\reverse\arrow\lines{(11,6),(11,3)}
\gfill[white]\tlabelrect[cc](11,4.5){$p$}
\end{mfpic}

Young�v diagram $\lambda^*=[4^2,3,1]$ reprezentace sdru�en� k~reprezentaci 
s~Youngov�m diagramem $\lambda=[4,3,1]$ pro $\dim V=n=5$.
\end{slide}
\begin{slide}
\section{M���kov� permutace}
Slovo $i_1 \cdots i_r$ ze symbol� $i_1,\ldots,i_r$, kde $1\le i_1,\ldots,i_r\le n$, nazveme m���kovou
permutac�, jestli�e ��dn� jeho podslovo $i_1\cdots i_s$ nebsahuje v�ce
$i+1$ ne� $i$ pro $1\le i \le n-1$.\pause

P��klad: 
\begin{enumerate}
\item[(1)] M���kov� permutace \emph{nen�} permutac�!
\item[(2)] Slovo $112321$ je m���kovou permutac�.
\item[(3)] Slovo $112331$ nen� m���kovou permutac�, proto�e
jeho podslovo $11233$ obsahuje v�cekr�t $3$ ne� $2$.\pause
\end{enumerate}

\emph{Clebsch-Gordanovy rozklady}

M���kov� permutace je t�eba uva�ovat p�i rozkladech tenzorov�ch sou�in�
ireducibiln�ch reprezentac� $[\lambda]\otimes [\mu]$ na ireducibiln�
reprezentace $[\nu]$.
\end{slide}
\begin{slide}
\section{Postup pro z�sk�n� ireducibiln� reprezetace}
Diagramy charakterizuj�c� reprezentace na $T^r_s V$ jsou d�ny dvojicemi
Youngov�ch diagram� $(\lambda,\mu)^r_s$. 
Je vhodn� dvojice zapisovat \uv{z�dy k sob�} a ozna�ovat diagram ur�uj�c�
kontravariantn� tenzory odli�n� nap�. te�kami, pro $([3,2,1],[2,1^3])$
$$
\young(\bullet\bullet\bullet\null\null,:\bullet\bullet\null,::\bullet\null,:::\null)
\pause
$$

Ireducibiln� reprezentace z�sk�me Clebsch-Gordanov�m rozkladem diagramu
$\lambda$ a diagramu sdru�en�ho k $\mu$. Form�ln� m��eme ps�t
$$
[\lambda]^r \otimes[\mu]_s = \sum_{\nu\rho} m^{\lambda\nu}_{\mu\rho}
[\nu;\rho]^{r-t}_{s-t}, \qquad \text{kde $0\le t\le \min(r,s)$.}
$$

Zb�v� ur�it rozklad $m^{\lambda\nu}_{\mu\rho}$ a k tomu
slou�� postup \cite{ki}.
\end{slide}
\begin{slide}
\section{Ur�en� rozkladu}
\begin{enumerate}
\item[(1)] Diagramy napi�me \uv{z�dy k sob�}, v diagramu pro kovariantn�
tenzory um�st�me te�ky.
\item[(2)] V pr�zdn�m diagramu vypln�me 1. ��dek symbolem 1, 2. ��dek
symbolem 2, \dots, $s$. ��dek symbolem $s$. 
\item[(3)] Pod ka�d� z $t$ symbol� um�st�me ��ru.
\item[(4)] V�echny symboly s �arou um�st�me m�sto te�ek v�emi mo�n�mi
zp�soby.\pause
\end{enumerate}

\emph{V ka�d�m kroku dodr�ujeme n�sleduj�c� pravidla:}
\begin{enumerate}
\item[(a)] V�sledn� obrazec op�t odpov�d� dvojici Youngov�ch diagram�,
pokud nahrad�me nep�epsan� te�ky a symboly bez �ar pr�zdn�mi �tverci.
\item[(b)] V�sledn� obrazec neobsahuje dva stejn� p�epsan� symboly v ��dn�m
sloupci.
\item[(c)] �teme-li ne��rkovan� symboly zprava doleva odshora dol� po ��dc�ch
diagramu $\mu$ dostaneme m���kovou permutaci.
\item[(d)] �teme-li symboly, kter�mi jsme p�epsali te�ky, zprava doleva zdola
nahoru po ��dc�ch diagramu $\lambda$ dostaneme m���kovou permutaci.
\end{enumerate}
\end{slide}
\begin{slide}
\section{Ur�en� rozkladu -- pokra�ov�n�}
\emph{Koeficient rozkladu $m^{\lambda\nu}_{\mu\rho}$ je po�et r�zn� o��slovan�ch
diagram� odpov�daj�c�ch diagramu $[\nu;\rho]^{r-t}_{s-t}$ z�skan�ch p�edchoz�m
postupem.}

D�le ozna�me $N$ po�et ��dk� diagramu $\nu$, $R$ po�et ��dk� diagramu $\rho$ a
$I$ ��slo nejv��e um�st�n�ho symbolu v prvn�m
sloupci diagramu $\lambda$. V�dy mus� platit
$$
N+R \le n=\dim V, \qquad\qquad
N+I \le n=\dim V.
$$
T�mito vzorci jsme schopni ur�it minim�ln� dimenzi $n_0$ vektorov�ho prostoru
$V$, pro ni� je reprezentace netrivi�ln�.
\end{slide}
\begin{slide}
\section{P��klad}
Rozklad $[3,2,1]\otimes[2^2]$.
\begin{gather*}
t=0, n_0=5:\\
\young(\bullet\bullet\bullet11,:\bullet\bullet22,::\bullet)\\
t=1, n_0=5:\\
\young(2\bullet\bullet11,:\bullet\bullet2\bD,::\bullet),\quad
\young(\bullet\bullet\bullet11,:2\bullet2\bD,::\bullet)\\
t=1, n_0=4:\\
\young(\bullet\bullet\bullet11,:\bullet\bullet2\bD,::2)\\
\end{gather*}
\end{slide}
\begin{slide}
\section{P��klad -- pokra�ov�n�}
\begin{gather*}
t=2, n_0=5:\\
\young(1\bullet\bullet1\bJ,:2\bullet2\bD,::\bullet)\quad
\young(2\bullet\bullet1\bJ,:1\bullet2\bD,::\bullet)\\
t=2, n_0=4:\\
\young(\bullet\bullet\bullet11,:2\bullet\bD\bD,::2)\quad
\young(2\bullet\bullet11,:\bullet\bullet\bD\bD,::2)\quad
\young(2\bullet\bullet11,:2\bullet\bD\bD,::\bullet)\\
\young(\bullet\bullet\bullet1\bJ,:1\bullet2\bD,::2)\quad
\young(\bullet\bullet\bullet1\bJ,:2\bullet2\bD,::1)\quad
\young(2\bullet\bullet1\bJ,:\bullet\bullet2\bD,::1)\quad
\young(1\bullet\bullet1\bJ,:\bullet\bullet2\bD,::2)\\
t=3, n_0=4:\\
\young(22\bullet1\bJ,:1\bullet\bD\bD,::\bullet)\quad
\young(21\bullet1\bJ,:2\bullet\bD\bD,::\bullet)\quad
\young(2\bullet\bullet1\bJ,:1\bullet\bD\bD,::2)
\end{gather*}
\end{slide}
\begin{slide}
\section{P��klad -- pokra�ov�n�}
\begin{gather*}
t=3, n_0=3:\\
\young(2\bullet\bullet1\bJ,:2\bullet\bD\bD,::1)\quad
\young(\bullet\bullet\bullet1\bJ,:22\bD\bD,::1)\\
t=3, n_0=2:\\
\young(\bullet\bullet\bullet1\bJ,:21\bD\bD,::2)\\
t=4, n_0=4:\\
\young(21\bullet\bJ\bJ,:1\bullet\bD\bD,::2)\\
t=4, n_0=3:\\
\young(22\bullet\bJ\bJ,:1\bullet\bD\bD,::1)\quad
\young(21\bullet\bJ\bJ,:2\bullet\bD\bD,::1)\quad
\young(2\bullet\bullet\bJ\bJ,:12\bD\bD,::1)
\end{gather*}
\end{slide}
\begin{slide}
\section{P��klad -- pokra�ov�n�}
\begin{gather*}
t=4, n_0=2:\\
\young(2\bullet\bullet\bJ\bJ,:21\bD\bD,::1)\quad
\young(2\bullet\bullet\bJ\bJ,:11\bD\bD,::2)
\end{gather*}
Celkem tedy dostaneme
\begin{multline*}
[3,2,1]^6\otimes[2^2]_4 = 1_5[3,2,1;2^2]^6_4+\\
+1_5[3,1^2;2,1]^5_3+1_5[2^2,1;2,1]^5_3+1_4[3,2;2,1]^5_3+\\
+2_5[2,1^2;1^2]^4_2+2_4[3,1;1^2]^4_2+2_4[2^2;1^2]^4_2+1_4[3,1;2]^4_2
+1_4[2^2;2]^4_2+1_4[2,1^2;2]^4_2+\\
+2_4[1^3;1]^3_1+2_{4+3}[2,1;1]^3_1+2_{3+2}[3;1]^3_1+\\
+3_{4+3+3}[1^2;\emptyset]^2_0+3_{3+2+2}[2;\emptyset]^2_0.
\end{multline*}
\end{slide}
\begin{slide}
\section{P��klad -- pokra�ov�n�}
\begin{multline*}
\young(\bullet\bullet\bullet,\bullet\bullet,\bullet)\otimes
\yng(2,2) =
1_5\young(\bullet\bullet\bullet\null\null,:\bullet\bullet\null\null,::\bullet)+\\
+1_5\young(\bullet\bullet\bullet\null\null,::\bullet\null,::\bullet)+
1_5\young(\bullet\bullet\null\null,\bullet\bullet\null,:\bullet)+
1_4\young(\bullet\bullet\bullet\null\null,:\bullet\bullet\null)+\\
\!\!\!\!\!\!\!\!\!\!
+2_5\young(\bullet\bullet\null,:\bullet\null,:\bullet)+
2_4\young(\bullet\bullet\bullet\null,::\bullet\null)+
2_4\young(\bullet\bullet\null,\bullet\bullet\null)+
1_4\young(\bullet\bullet\bullet\null\null,::\bullet)+
1_4\young(\bullet\bullet\null\null,\bullet\bullet)+
1_4\young(\bullet\bullet\null\null,:\bullet,:\bullet)+\\
+2_4\young(\bullet\null,\bullet,\bullet)+
+2_{4+3}\young(\bullet\bullet\null,:\bullet)+
2_{3+2}\young(\bullet\bullet\bullet\null)+\\
+3_{4+3+3}\young(\bullet,\bullet)+3_{3+2+2}\young(\bullet\bullet).
\end{multline*}
\end{slide}
\begin{slide}
\section{Pro kter� ��seln� t�lesa m��eme rozklad prov�d�t?}
\begin{itemize}
\item Reprezentace grup $\SM_k$ a $\GL(n)$ mus� b�t ve vektorov�ch prostorech
nad t�lesem $\K$ \emph{�pln� reducibiln�}.\pause
\item Pro kone�n� grupy je rozhodnut� o �pln� reducibilit� jednoduch�: Po�et
prvk� grupy nesm� b�t d�liteln� charakteristikou t�lesa, specieln� pro $\SM_k$
$$
\Char \K > k.\pause
$$
\item Je-li $\SM_k$ �pln� reducibiln�, potom i k n� koresponduj�c� grupa je
�pln� reducibiln� (V�ta o dvojit�m centraliz�ru).\pause
\item Korespondence mezi $\SM_k$ a $\GL(n)$ plat� pro $\Char \K= 0$ a
pro $\Char\K\ne 0$ koresponduje k $\SM_k$ obecn� jin� grupa.\pause
\item Celkem tedy uveden� postupy plat� pro $\Char\K=0$, tzn. nap�. komplexn�,
re�ln� a racion�ln� ��sla (detaily \cite{kp}).
\end{itemize}
\end{slide}
\closegraphsfile
\end{document}
