excited - 山梨大学hnaruse/ken/2011.09.28eyd-yb.pdfk-theor y) n 5 %? 7 $ n 5 %? 7 $ (k-theor y h)...
TRANSCRIPT
2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 1 / 10
Ex ited Young diagram Yang-Baxter
�
� � # G � 7 # � �
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 2 / 10
� = (3; 2) � � � �� � �� � x1 x1 x1 x1 x1 x1x2 x2 x2 x2 x2 x2x3 x3 x3 x3 x3 x3x4 x4 x4 x4 x4 x4
x31x22+ x21x32+ x21x22x3+ x21x22x4+ x21x2x3x4+ x1x22x3x4+
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 2 / 10
� = (3; 2) � � � �� � �� � x1 x1 x1 x1 x1 x1x2 x2 x2 x2 x2 x2x3 x3 x3 x3 x3 x3x4 x4 x4 x4 x4 x4
D = � � �� � xw(D) = x31x22
x31x22+
x21x32+ x21x22x3+ x21x22x4+ x21x2x3x4+ x1x22x3x4+
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 2 / 10
� = (3; 2) � � � �� � �� � x1 x1 x1 x1 x1 x1x2 x2 x2 x2 x2 x2x3 x3 x3 x3 x3 x3x4 x4 x4 x4 x4 x4
D = � �� � � xw(D) = x21x32
x31x22+ x21x32+
x21x22x3+ x21x22x4+ x21x2x3x4+ x1x22x3x4+
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 2 / 10
� = (3; 2) � � � �� � �� � x1 x1 x1 x1 x1 x1x2 x2 x2 x2 x2 x2x3 x3 x3 x3 x3 x3x4 x4 x4 x4 x4 x4
D = � �� �� xw(D) = x21x22x3
x31x22+ x21x32+ x21x22x3+
x21x22x4+ x21x2x3x4+ x1x22x3x4+
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 2 / 10
� = (3; 2) � � � �� � �� � x1 x1 x1 x1 x1 x1x2 x2 x2 x2 x2 x2x3 x3 x3 x3 x3 x3x4 x4 x4 x4 x4 x4
D = � �� �� xw(D) = x21x22x4
x31x22+ x21x32+ x21x22x3+ x21x22x4+
x21x2x3x4+ x1x22x3x4+
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 2 / 10
� = (3; 2) � � � �� � �� � x1 x1 x1 x1 x1 x1x2 x2 x2 x2 x2 x2x3 x3 x3 x3 x3 x3x4 x4 x4 x4 x4 x4
D = � � �� � xw(D) = x21x2x3x4
x31x22+ x21x32+ x21x22x3+ x21x22x4+ x21x2x3x4+
x1x22x3x4+
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 2 / 10
� = (3; 2) � � � �� � �� � x1 x1 x1 x1 x1 x1x2 x2 x2 x2 x2 x2x3 x3 x3 x3 x3 x3x4 x4 x4 x4 x4 x4
D = � � �� � xw(D) = x1x22x3x4
x31x22+ x21x32+ x21x22x3+ x21x22x4+ x21x2x3x4+ x1x22x3x4+
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 2 / 10
� = (3; 2) � � � �� � �� � x1 x1 x1 x1 x1 x1x2 x2 x2 x2 x2 x2x3 x3 x3 x3 x3 x3x4 x4 x4 x4 x4 x4
x31x22+ x21x32+ x21x22x3+ x21x22x4+ x21x2x3x4+ x1x22x3x4+
% � ! � 4 ?s32(x1; x2; x3; x4)= x31x22 + x21x32 + x21x22x3 + x21x22x4 + x21x2x3x4 + x1x22x3x4 +� � �+ x23x34 s�(x1; :::; xn) = XD2En(�)xw(D)
(K-theory )� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 3 / 10
Grothendie k polynomial G�(x1; :::; xn) := XD2En(�)Wt�(D);Wt�(D) := xw(D)Qb2B(D)(1 + �xw(b))� � �� � x1 x1 x1 x1 x1 x1x2 x2 x2 x2 x2 x2x3 x3 x3 x3 x3 x3x4 x4 x4 x4 x4 x4
(K-theory )� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 3 / 10
Grothendie k polynomial G�(x1; :::; xn) := XD2En(�)Wt�(D);Wt�(D) := xw(D)Qb2B(D)(1 + �xw(b))� � �� � x1 x1 x1 x1 x1 x1x2 x2 x2 x2 x2 x2x3 x3 x3 x3 x3 x3x4 x4 x4 x4 x4 x4�� � � � x1x22x24
(K-theory )� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 3 / 10
Grothendie k polynomial G�(x1; :::; xn) := XD2En(�)Wt�(D);Wt�(D) := xw(D)Qb2B(D)(1 + �xw(b))� � �� � x1 x1 x1 x1 x1 x1x2 x2 x2 x2 x2 x2x3 x3 x3 x3 x3 x3x4 x4 x4 x4 x4 x4� b� � bb b� �
x1x22x24(1 + �x1)(1 + �x2)(1 + �x3)2
(K-theory )� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 3 / 10
Grothendie k polynomial G�(x1; :::; xn) := XD2En(�)Wt�(D);Wt�(D) := xw(D)Qb2B(D)(1 + �xw(b))� � �� � x1 x1 x1 x1 x1 x1x2 x2 x2 x2 x2 x2x3 x3 x3 x3 x3 x3x4 x4 x4 x4 x4 x4
% � � � � � � $ 2 5 � � � � � � � fa torial Grothendie kpolynomial � ! � � � � x� y = x+ y + �xyx1 � b1 x1 � b2 x1 � b3 x1 � b4 x1 � b5 x1 � b6x2 � b1 x2 � b2 x2 � b3 x2 � b4 x2 � b5 x2 � b6x3 � b1 x3 � b2 x3 � b3 x3 � b4 x3 � b5 x3 � b6x4 � b1 x4 � b2 x4 � b3 x4 � b4 x4 � b5 x4 � b6
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 4 / 10
x� b1 x� b2 x� b3 x� b4 x� b5 00 y � b2 y � b3 y � b4 y � b5 y � b6my � b1 y � b2 y � b3 y � b4 y � b5 00 x� b2 x� b3 x� b4 x� b5 x� b6
a� b1 a� b2 a� b3 a� b4 a� b5 00 b2 b3 b4 b5 b6mb1 b2 b3 b4 b5 00 a� b2 a� b3 a� b4 a� b5 a� b6a = x y, bi = y � bix y = x�y1+�y
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 4 / 10
x� b1 x� b2 x� b3 x� b4 x� b5 00 y � b2 y � b3 y � b4 y � b5 y � b6my � b1 y � b2 y � b3 y � b4 y � b5 00 x� b2 x� b3 x� b4 x� b5 x� b6
a� b1 a� b2 a� b3 a� b4 a� b5 00 b2 b3 b4 b5 b6mb1 b2 b3 b4 b5 00 a� b2 a� b3 a� b4 a� b5 a� b6
a = x y, bi = y � bix y = x�y1+�y
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 4 / 10
x� b1 x� b2 x� b3 x� b4 x� b5 00 y � b2 y � b3 y � b4 y � b5 y � b6my � b1 y � b2 y � b3 y � b4 y � b5 00 x� b2 x� b3 x� b4 x� b5 x� b6
a� b1 a� b2 a� b3 a� b4 a� b5 00 b2 b3 b4 b5 b6mb1 b2 b3 b4 b5 00 a� b2 a� b3 a� b4 a� b5 a� b6a = x y, bi = y � bi � � $ � � > � � 9 � 0 C � !
� � � � x y = x�y1+�y
Id-Coxeter Yang-Baxter� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 5 / 10
@ � 8 u0; u1; :::, I D 8 �, 4 7 < $ � � � � � !u2i = �uiuiuj = ujui (ji� jj � 2)uiui+1ui = ui+1uiui+1 (i � 1)u0u1u0u1 = u1u0u1u0
hi(x) = 1 + xuihi(x)hi(y) = hi(y)hi(x) = hi(x� y)hi(x)hj(y) = hj(y)hi(x) (ji� jj � 2)hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)h0(x)h1(x� y)h0(x� y � y)h1(y)= h1(y)h0(x� y � y)h1(x� y)h0(x)
Id-Coxeter Yang-Baxter� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 5 / 10
@ � 8 u0; u1; :::, I D 8 �, 4 7 < $ � � � � � !u2i = �uiuiuj = ujui (ji� jj � 2)uiui+1ui = ui+1uiui+1 (i � 1)u0u1u0u1 = u1u0u1u0
hi(x) = 1 + xui � $ � � �hi(x)hi(y) = hi(y)hi(x) = hi(x� y)hi(x)hj(y) = hj(y)hi(x) (ji� jj � 2)hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)h0(x)h1(x� y)h0(x� y � y)h1(y)= h1(y)h0(x� y � y)h1(x� y)h0(x)
Yang-Baxter� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 6 / 10
hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)i& i+ 1&x x� y 00 y 0 () i& i+ 1&0 y 00 x� y x
h0(x)h1(x� y)h0(x� y � y)h1(y)= h1(y)h0(x� y � y)h1(x� y)h0(x)0& 1& 0& 1&x x� y 0 0 y 0x� y � y y () x� y � y x� y0 x
Yang-Baxter� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 6 / 10
hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)i& i+ 1&x x� y 00 y 0 () i& i+ 1&0 y 00 x� y x
h0(x)h1(x� y)h0(x� y � y)h1(y)= h1(y)h0(x� y � y)h1(x� y)h0(x)0& 1& 0& 1&x x� y 0 0 y 0x� y � y y () x� y � y x� y0 x
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 7 / 10
hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)i& i+ 1&x x� y 00 y 0 () i& i+ 1&0 y 00 x� y x
a� b1 a� b2 a� b3 a� b4 a� b5 00 b2 b3 b4 b5 b6
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 7 / 10
hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)i& i+ 1&x x� y 00 y 0 () i& i+ 1&0 y 00 x� y x
a� b1 a� b2 a� b3 a� b4 a� b5 00 b2 b3 b4 b5 b6
b1 00 a a� b2 00 b2 0 a� b3 a� b4 a� b5 00 b3 b4 b5 b6
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 7 / 10
hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)i& i+ 1&x x� y 00 y 0 () i& i+ 1&0 y 00 x� y x
a� b1 a� b2 a� b3 a� b4 a� b5 00 b2 b3 b4 b5 b6
b1 00 b2 00 a� b2 a a� b3 a� b4 a� b5 00 b3 b4 b5 b6
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 7 / 10
hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)i& i+ 1&x x� y 00 y 0 () i& i+ 1&0 y 00 x� y x
a� b1 a� b2 a� b3 a� b4 a� b5 00 b2 b3 b4 b5 b6
b1 b2 00 a� b2 a a� b3 00 b3 0 a� b4 a� b5 00 b4 b5 b6
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 7 / 10
hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)i& i+ 1&x x� y 00 y 0 () i& i+ 1&0 y 00 x� y x
a� b1 a� b2 a� b3 a� b4 a� b5 00 b2 b3 b4 b5 b6
b1 b2 00 a� b2 0 b3 00 a� b3 a a� b4 a� b5 00 b4 b5 b6
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 7 / 10
hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)i& i+ 1&x x� y 00 y 0 () i& i+ 1&0 y 00 x� y x
a� b1 a� b2 a� b3 a� b4 a� b5 00 b2 b3 b4 b5 b6
b1 b2 b3 00 a� b2 a� b3 a a� b4 00 b4 0 a� b5 00 b5 b6
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 7 / 10
hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)i& i+ 1&x x� y 00 y 0 () i& i+ 1&0 y 00 x� y x
a� b1 a� b2 a� b3 a� b4 a� b5 00 b2 b3 b4 b5 b6
b1 b2 b3 00 a� b2 a� b3 0 b4 00 a� b4 a a� b5 00 b5 b6
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 7 / 10
hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)i& i+ 1&x x� y 00 y 0 () i& i+ 1&0 y 00 x� y x
a� b1 a� b2 a� b3 a� b4 a� b5 00 b2 b3 b4 b5 b6
b1 b2 b3 b4 00 a� b2 a� b3 a� b4 a a� b5 00 b5 0 00 b6
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 7 / 10
hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)i& i+ 1&x x� y 00 y 0 () i& i+ 1&0 y 00 x� y x
a� b1 a� b2 a� b3 a� b4 a� b5 00 b2 b3 b4 b5 b6
b1 b2 b3 b4 00 a� b2 a� b3 a� b4 0 b5 00 a� b5 a 00 b6
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 7 / 10
hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)i& i+ 1&x x� y 00 y 0 () i& i+ 1&0 y 00 x� y x
a� b1 a� b2 a� b3 a� b4 a� b5 00 b2 b3 b4 b5 b6
b1 b2 b3 b4 b5 00 a� b2 a� b3 a� b4 a� b5 a 00 b6
� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 7 / 10
hi(x)hi+1(x� y)hi(y) = hi+1(y)hi(x� y)hi+1(x) (i � 1)i& i+ 1&x x� y 00 y 0 () i& i+ 1&0 y 00 x� y x
a� b1 a� b2 a� b3 a� b4 a� b5 00 b2 b3 b4 b5 b6mb1 b2 b3 b4 b5 00 a� b2 a� b3 a� b4 a� b5 a� b6
skew shape� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 8 / 10
� = (2; 2; 1) � � = (1), skew % � ! � 4 ? s�=�(x)�� ��x1 x1 x1 x1 x1x2 x2 x2 x2 x2x3 x3 x3 x3 x3x4 x4 x4 x4 x4
D = �� ��w(D) = x1x22x3D = � ���w(D) = x21x2x3s(2;2;1)=(1)(x) = s(1;1)(x)2 � s(1;1;1;1)(x)
skew shape� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 8 / 10
� = (2; 2; 1) � � = (1), skew % � ! � 4 ? s�=�(x)�� ��x1 x1 x1 x1 x1x2 x2 x2 x2 x2x3 x3 x3 x3 x3x4 x4 x4 x4 x4
D = �� ��w(D) = x1x22x3D = � ���w(D) = x21x2x3
s(2;2;1)=(1)(x) = s(1;1)(x)2 � s(1;1;1;1)(x)
skew shape� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 8 / 10
� = (2; 2; 1) � � = (1), skew % � ! � 4 ? s�=�(x)�� ��x1 x1 x1 x1 x1x2 x2 x2 x2 x2x3 x3 x3 x3 x3x4 x4 x4 x4 x4
D = �� ��w(D) = x1x22x3D = � ���w(D) = x21x2x3s(2;2;1)=(1)(x) = s(1;1)(x)2 � s(1;1;1;1)(x)
oprodu t� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 9 / 10
... ... ...x1 x2 x3x1 x2 x3x1 � x1 x1 � x2 x1 � x3 x1 x1 x1 � � �x2 � x2 x2 � x3 x2 x2 x2 � � �x3 � x3 x3 x3 x3 � � �
�=� shifted skew diagram, e.g. (3; 2)=(2) = �� �GQ�=�(x) := XD2E(�=�)xw(D) Yb2B(D)(1 + �xw(b))
�(GQ�=�(x)) = X�=�0 rook�����0���j�=�0jGQ�=�0(x)GQ�=�(x)
oprodu t� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 9 / 10
... ... ...x1 x2 x3x1 x2 x3x1 � x1 x1 � x2 x1 � x3 x1 x1 x1 � � �x2 � x2 x2 � x3 x2 x2 x2 � � �x3 � x3 x3 x3 x3 � � �
�=� shifted skew diagram, e.g. (3; 2)=(2) = �� �GQ�=�(x) := XD2E(�=�)xw(D) Yb2B(D)(1 + �xw(b))
�(GQ�=�(x)) = X�=�0 rook�����0���j�=�0jGQ�=�0(x)GQ�=�(x)
pipedream� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 10 / 10
Pipe dream formula for Cn, S hubertCnw = XD2E(Ln;w)Wt(D)
L3 = x1 � b2x1 � b1 x2 � b1x1 � x1 x1 � x2 x1 � x3x2 � x2 x2 � x3 x2 � a1x3 � x3 x3 � a1 x3 � a221 20 1 20 1 20 1 2S hubertC3s1= x1 � x1 � x2 � x2 � x3 � x3 � a1 � b1
pipedream� N 5 % � � ? 7
$ �� N 5 % � � ? 7 $
� (K-theory H )� � � � � � F C J 7� Id-Coxeter B � $Yang-Baxter 4 <� K J 7 � � ! �
$ Yang-Baxter 4 <� K J 7 � F C
J 7 �� skew shape� oprodu t� pipedream2011.09.28 F K ? 3 2 = 5 A 9 J � 2 0 H � 1 � ; B 3 B 3 � 6 � 3 8 � � @ � � � 10 / 10
Pipe dream formula for Cn, S hubertCnw = XD2E(Ln;w)Wt(D)
L3 = x1 � b2x1 � b1 x2 � b1x1 � x1 x1 � x2 x1 � x3x2 � x2 x2 � x3 x2 � a1x3 � x3 x3 � a1 x3 � a221 20 1 20 1 20 1 2S hubertC3s1= x1 � x1 � x2 � x2 � x3 � x3 � a1 � b1
Formula for w 2 Sn �W (Cn), S hubertCnw (x; z; t)= S hubertA2n1n�w(x1; :::; xn; z1; :::; zn�1;x1; :::; xn; t1; :::; tn�1)