Hall algebra - 편집 역사

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imported>Pythagoras0: 새 문서: ==expositions== * Dyckerhoff, Tobias. ‘Higher Categorical Aspects of Hall Algebras’. arXiv:1505.06940 [math], 26 May 2015. http://arxiv.org/abs/1505.06940.

2015-05-27T03:50:09Z

새 문서: ==expositions== * Dyckerhoff, Tobias. ‘Higher Categorical Aspects of Hall Algebras’. arXiv:1505.06940 [math], 26 May 2015. http://arxiv.org/abs/1505.06940.

새 문서

==expositions==
* Dyckerhoff, Tobias. ‘Higher Categorical Aspects of Hall Algebras’. arXiv:1505.06940 [math], 26 May 2015. http://arxiv.org/abs/1505.06940.

@@ 46번째 줄: / 46번째 줄: @@
 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q5642657 Q5642657]

← 이전 판		2020년 12월 28일 (월) 06:48 판
45번째 줄:		45번째 줄:
	* Scherotzke, Sarah, and Nicolo Sibilla. “Quiver Varieties and Hall Algebras.” arXiv:1506.03609 [math], June 11, 2015. http://arxiv.org/abs/1506.03609.		* Scherotzke, Sarah, and Nicolo Sibilla. “Quiver Varieties and Hall Algebras.” arXiv:1506.03609 [math], June 11, 2015. http://arxiv.org/abs/1506.03609.
	[[분류:migrate]]		[[분류:migrate]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q5642657 Q5642657]

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
+<math>
 \newcommand{\la}{\lambda}
 \newcommand{\La}{\Lambda}
+</math>
 ==Hall polynomials==
-Recall that the Littlewood-Richardson coefficient $c^{\la}_{\mu \nu}$ is equal to the number of tableaux $T$ of shape $\la - \mu$ and weight $\nu$ such that $w(T)$, the word of $T$, is a lattice permutation.  We have
+Recall that the Littlewood-Richardson coefficient <math>c^{\la}_{\mu \nu}</math> is equal to the number of tableaux <math>T</math> of shape <math>\la - \mu</math> and weight <math>\nu</math> such that <math>w(T)</math>, the word of <math>T</math>, is a lattice permutation.  We have
 \begin{equation*}
 s_{\mu}s_{\nu} = \sum_{\la} c^{\la}_{\mu \nu} s_{\la},
 \end{equation*}
-where $s_{\mu}$ is the Schur function.
+where <math>s_{\mu}</math> is the Schur function.
-We briefly recall the Hall polynomials $g_{\mu \nu}^{\la}(q)$ \cite[Chs. II and V]{Mac}.  Let $\mathcal{O}$ be a complete (commutative) discrete valuation ring, $\mathcal{P}$ its maximal ideal and $k = \mathcal{O}/\mathcal{P}$ the residue field.  We assume $k$ is a finite field.  Let $q$ be the number of elements in $k$.  Let $M$ be a finite $\mathcal{O}$-module of type $\la$.  Then the number of submodules of $N$ of $M$ with type $\nu$ and cotype $\mu$ is a polynomial in $q$, called the Hall polynomial, denoted $g_{\mu \nu}^{\la}(q)$.  One can consider our motivating case of $\mathbb{Q}_{p}$ and its ring of integers $\mathcal{O} = \mathbb{Z}_{p}$ and $G = Gl_{n}(\mathbb{Q}_{p})$, so that $q=p$.  Then they are also the structure constants for the ring $\mathcal{H}(G^{+},K)$.  That is, for $\mu, \nu \in \La_{2n}^{+}$, we have
+We briefly recall the Hall polynomials <math>g_{\mu \nu}^{\la}(q)</math> \cite[Chs. II and V]{Mac}.  Let <math>\mathcal{O}</math> be a complete (commutative) discrete valuation ring, <math>\mathcal{P}</math> its maximal ideal and <math>k = \mathcal{O}/\mathcal{P}</math> the residue field.  We assume <math>k</math> is a finite field.  Let <math>q</math> be the number of elements in <math>k</math>.  Let <math>M</math> be a finite <math>\mathcal{O}</math>-module of type <math>\la</math>.  Then the number of submodules of <math>N</math> of <math>M</math> with type <math>\nu</math> and cotype <math>\mu</math> is a polynomial in <math>q</math>, called the Hall polynomial, denoted <math>g_{\mu \nu}^{\la}(q)</math>.  One can consider our motivating case of <math>\mathbb{Q}_{p}</math> and its ring of integers <math>\mathcal{O} = \mathbb{Z}_{p}</math> and <math>G = Gl_{n}(\mathbb{Q}_{p})</math>, so that <math>q=p</math>.  Then they are also the structure constants for the ring <math>\mathcal{H}(G^{+},K)</math>.  That is, for <math>\mu, \nu \in \La_{2n}^{+}</math>, we have
 \begin{equation*}
 c_{\mu} \star c_{\nu} = \sum_{\la \in \La_{2n}^{+}} g_{\mu \nu}^{\la}(p) c_{\la}.
@@ 22번째 줄: / 22번째 줄: @@
 Several important facts are known (see \cite[Ch. II]{Mac}):
-* If $c^{\la}_{\mu \nu} = 0$, then $g^{\la}_{\mu \nu}(t) = 0$ as a function of $t$.
+* If <math>c^{\la}_{\mu \nu} = 0</math>, then <math>g^{\la}_{\mu \nu}(t) = 0</math> as a function of <math>t</math>.
-* If $c^{\la}_{\mu \nu} \neq 0$, then $g^{\la}_{\mu \nu}(t)$ has degree $n(\la) - n(\mu) - n(\nu)$ and leading coefficient $c^{\la}_{\mu \nu}$, where the notation $n(\la) = \sum (i-1) \la_{i}$.
+* If <math>c^{\la}_{\mu \nu} \neq 0</math>, then <math>g^{\la}_{\mu \nu}(t)</math> has degree <math>n(\la) - n(\mu) - n(\nu)</math> and leading coefficient <math>c^{\la}_{\mu \nu}</math>, where the notation <math>n(\la) = \sum (i-1) \la_{i}</math>.
-* We have $g^{\la}_{\mu \nu}(t) = g^{\la}_{\nu \mu}(t)$.
+* We have <math>g^{\la}_{\mu \nu}(t) = g^{\la}_{\nu \mu}(t)</math>.
@@ 31번째 줄: / 31번째 줄: @@
 P_{\mu}(x;t) P_{\nu}(x;t) = \sum_{\la} f^{\la}_{\mu \nu}(t) P_{\la}(x;t),
 \end{equation*}
-with $f^{\la}_{\mu \nu}(t) = t^{n(\la) - n(\mu) - n(\nu)} g^{\la}_{\mu \nu}(t^{-1})$.
+with <math>f^{\la}_{\mu \nu}(t) = t^{n(\la) - n(\mu) - n(\nu)} g^{\la}_{\mu \nu}(t^{-1})</math>.

← 이전 판		2020년 11월 13일 (금) 14:43 판
44번째 줄:		44번째 줄:
	==articles==		==articles==
	* Scherotzke, Sarah, and Nicolo Sibilla. “Quiver Varieties and Hall Algebras.” arXiv:1506.03609 [math], June 11, 2015. http://arxiv.org/abs/1506.03609.		* Scherotzke, Sarah, and Nicolo Sibilla. “Quiver Varieties and Hall Algebras.” arXiv:1506.03609 [math], June 11, 2015. http://arxiv.org/abs/1506.03609.
		+	[[분류:migrate]]

← 이전 판		2019년 11월 21일 (목) 02:37 판
1번째 줄:		1번째 줄:
		+	==introduction==
		+	$
		+	\newcommand{\la}{\lambda}
		+	\newcommand{\La}{\Lambda}
		+	$
		+
		+	==Hall polynomials==
		+	Recall that the Littlewood-Richardson coefficient $c^{\la}_{\mu \nu}$ is equal to the number of tableaux $T$ of shape $\la - \mu$ and weight $\nu$ such that $w(T)$, the word of $T$, is a lattice permutation. We have
		+	\begin{equation*}
		+	s_{\mu}s_{\nu} = \sum_{\la} c^{\la}_{\mu \nu} s_{\la},
		+	\end{equation*}
		+	where $s_{\mu}$ is the Schur function.
		+
		+	We briefly recall the Hall polynomials $g_{\mu \nu}^{\la}(q)$ \cite[Chs. II and V]{Mac}. Let $\mathcal{O}$ be a complete (commutative) discrete valuation ring, $\mathcal{P}$ its maximal ideal and $k = \mathcal{O}/\mathcal{P}$ the residue field. We assume $k$ is a finite field. Let $q$ be the number of elements in $k$. Let $M$ be a finite $\mathcal{O}$-module of type $\la$. Then the number of submodules of $N$ of $M$ with type $\nu$ and cotype $\mu$ is a polynomial in $q$, called the Hall polynomial, denoted $g_{\mu \nu}^{\la}(q)$. One can consider our motivating case of $\mathbb{Q}_{p}$ and its ring of integers $\mathcal{O} = \mathbb{Z}_{p}$ and $G = Gl_{n}(\mathbb{Q}_{p})$, so that $q=p$. Then they are also the structure constants for the ring $\mathcal{H}(G^{+},K)$. That is, for $\mu, \nu \in \La_{2n}^{+}$, we have
		+	\begin{equation*}
		+	c_{\mu} \star c_{\nu} = \sum_{\la \in \La_{2n}^{+}} g_{\mu \nu}^{\la}(p) c_{\la}.
		+	\end{equation*}
		+	Note that, in particular,
		+	\begin{equation*}
		+	g_{\mu \nu}^{\la}(p) = (c_{\mu} \star c_{\nu})(p^{\la}) = \int_{G} c_{\mu}(p^{\la}y^{-1})c_{\nu}(y)dy = meas.(p^{\la}Kp^{-\nu}K \cap Kp^{\mu}K).
		+	\end{equation*}
		+
		+	Several important facts are known (see \cite[Ch. II]{Mac}):
		+	* If $c^{\la}_{\mu \nu} = 0$, then $g^{\la}_{\mu \nu}(t) = 0$ as a function of $t$.
		+	* If $c^{\la}_{\mu \nu} \neq 0$, then $g^{\la}_{\mu \nu}(t)$ has degree $n(\la) - n(\mu) - n(\nu)$ and leading coefficient $c^{\la}_{\mu \nu}$, where the notation $n(\la) = \sum (i-1) \la_{i}$.
		+	* We have $g^{\la}_{\mu \nu}(t) = g^{\la}_{\nu \mu}(t)$.
		+
		+
		+	Also if one multiplies two Hall-Littlewood polynomials, and expands the result in the Hall-Littlewood basis, one has
		+	\begin{equation*}
		+	P_{\mu}(x;t) P_{\nu}(x;t) = \sum_{\la} f^{\la}_{\mu \nu}(t) P_{\la}(x;t),
		+	\end{equation*}
		+	with $f^{\la}_{\mu \nu}(t) = t^{n(\la) - n(\mu) - n(\nu)} g^{\la}_{\mu \nu}(t^{-1})$.
		+
		+
		+	==related items==
		+	* {{수학노트\|url=홀-리틀우드(Hall-Littlewood)_대칭함수}}
		+
		+
	==expositions==		==expositions==
	* Dyckerhoff, Tobias. ‘Higher Categorical Aspects of Hall Algebras’. arXiv:1505.06940 [math], 26 May 2015. http://arxiv.org/abs/1505.06940.		* Dyckerhoff, Tobias. ‘Higher Categorical Aspects of Hall Algebras’. arXiv:1505.06940 [math], 26 May 2015. http://arxiv.org/abs/1505.06940.

← 이전 판		2015년 6월 12일 (금) 05:20 판
1번째 줄:		1번째 줄:
	==expositions==		==expositions==
	* Dyckerhoff, Tobias. ‘Higher Categorical Aspects of Hall Algebras’. arXiv:1505.06940 [math], 26 May 2015. http://arxiv.org/abs/1505.06940.		* Dyckerhoff, Tobias. ‘Higher Categorical Aspects of Hall Algebras’. arXiv:1505.06940 [math], 26 May 2015. http://arxiv.org/abs/1505.06940.
		+
		+
		+	==articles==
		+	* Scherotzke, Sarah, and Nicolo Sibilla. “Quiver Varieties and Hall Algebras.” arXiv:1506.03609 [math], June 11, 2015. http://arxiv.org/abs/1506.03609.