8 results
23 - Equivariant Cohomology, Schubert Calculus, and Edge Labeled Tableaux
- Edited by Paolo Aluffi, Florida State University, David Anderson, Ohio State University, Milena Hering, University of Edinburgh, Mircea Mustaţă, University of Michigan, Ann Arbor, Sam Payne, University of Texas, Austin
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- Facets of Algebraic Geometry
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- 14 March 2022
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- 07 April 2022, pp 284-335
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EQUIVARIANT $K$-THEORY OF GRASSMANNIANS
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- OLIVER PECHENIK, ALEXANDER YONG
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- Forum of Mathematics, Pi / Volume 5 / 2017
- Published online by Cambridge University Press:
- 27 June 2017, e3
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We address a unification of the Schubert calculus problems solved by Buch [A Littlewood–Richardson rule for the $K$-theory of Grassmannians, Acta Math. 189 (2002), 37–78] and Knutson and Tao [Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J.119(2) (2003), 221–260]. That is, we prove a combinatorial rule for the structure coefficients in the torus-equivariant $K$-theory of Grassmannians with respect to the basis of Schubert structure sheaves. This rule is positive in the sense of Anderson et al. [Positivity and Kleiman transversality in equivariant $K$-theory of homogeneous spaces, J. Eur. Math. Soc.13 (2011), 57–84] and in a stronger form. Our work is based on the combinatorics of genomic tableaux and a generalization of Schützenberger’s [Combinatoire et représentation du groupe symétrique, in Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976, Lecture Notes in Mathematics, 579 (Springer, Berlin, 1977), 59–113] jeu de taquin. Using our rule, we deduce the two other combinatorial rules for these coefficients. The first is a conjecture of Thomas and Yong [Equivariant Schubert calculus and jeu de taquin, Ann. Inst. Fourier (Grenoble) (2013), to appear]. The second (found in a sequel to this paper) is a puzzle rule, resolving a conjecture of Knutson and Vakil from 2005.
Equivariant $K$-theory of Grassmannians II: the Knutson–Vakil conjecture
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- Oliver Pechenik, Alexander Yong
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- Journal:
- Compositio Mathematica / Volume 153 / Issue 4 / April 2017
- Published online by Cambridge University Press:
- 09 March 2017, pp. 667-677
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- April 2017
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In 2005, Knutson–Vakil conjectured a puzzle rule for equivariant $K$-theory of Grassmannians. We resolve this conjecture. After giving a correction, we establish a modified rule by combinatorially connecting it to the authors’ recently proved tableau rule for the same Schubert calculus problem.
Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians
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- David Anderson, Edward Richmond, Alexander Yong
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- Journal:
- Compositio Mathematica / Volume 149 / Issue 9 / September 2013
- Published online by Cambridge University Press:
- 07 August 2013, pp. 1569-1582
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- September 2013
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The saturation theorem of Knutson and Tao concerns the nonvanishing of Littlewood–Richardson coefficients. In combination with work of Klyachko, it implies Horn’s conjecture about eigenvalues of sums of Hermitian matrices. This eigenvalue problem has a generalization to majorized sums of Hermitian matrices, due to S. Friedland. We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland’s problem, via equivariant cohomology of Grassmannians. In particular, we prove a saturation theorem for this setting. Our arguments employ the aforementioned work together with recent work of H. Thomas and A. Yong.
Contributors
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- By Luis G. Acevedo, Schahram Akbarian, Ioanna Andreou, Krishnarao Appasani, Raghu K. Appasani, Julia Arand, David M. Ashley, Alexander R. Ball, Yehudit Bergman, Marina Bibikova, Angela Bithell, Francesca Bonafè, Eric E. Bouhassira, Victoria L. Boyd, Noel J. Buckley, Lars Olov Bygren, Claudio M. Caldarera, Gemma Carvill, James W. F. Catto, Sarah Derks, Ewa Dudziec, Jeffrey D. Falk, Jian-Bing Fan, Joseph M. Fernandez, David E. Fisher, Emanuela Fiumana, Tamara B. Franklin, Fei Gao, Arkadiusz Gertych, Emanuele Giordano, David Goldman, Markus Grammel, Carlo Guarnieri, Kevin L. Gunderson, Victoria (Fatemeh) G. Haghighi, Xu Han, Yong-Mahn Han, Howard C. Hang, Aditi Hazra, Laura B.K. Herzing, Norbert Hochstein, Robin Holliday, Dorothee Honsel, Mary A. Jelinek, Guanyu Ji, Yan Jiang, Atsushi Kaneda, Richard A. Katz, Hyemin Kim, Richard Kroon, Tapas K. Kundu, Benoit Labonté, Daeyoup Lee, Konstantin Lepikhov, Andrea Linnemann-Florl, Dirk Loeffert, Dylan Maixner, Isabelle M. Mansuy, Andreas Missel, D. V. Mohankrishna, Joana Carvalho Moreira de Mello, Paolo G. Morselli, Rituparna Mukhopadhyay, Claudio Muscari, Takashi Nagano, Frank Narz, Shuji Ogino, Carlo M. Oranges, Shari Orlanski, Alice Pasini, Ralf Peist, Lygia V. Pereira, Andrey Poleshko, Claire Rougeulle, Thea Rütjes, Ana Sanz, Benjamin G. Schroeder, Gerald Schock, Kornel Schuebel, B. Ruthrotha Selvi, Hogyu Seo, Natalia Shalginskikh, Andrew Sharp, Jun S. Song, Lennart Suckau, Azim Surani, Jian Tajbakhsh, Gustavo Turecki, Céline Vallot, Manon van Engeland, Jörn Walter, Nicholas C. Wong, Mark Wossidlo, Honglong Wu, Yurong Xin, Zhixiang Yan, Yu-Ying Yang, Mingzhi Ye, Kyoko Yokomori, Sephorah Zaman, Weihua Zeng, Gerald Zon
- Edited by Krishnarao Appasani
- Foreword by Azim Surani, University of Cambridge
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- Book:
- Epigenomics
- Published online:
- 05 August 2012
- Print publication:
- 02 August 2012, pp x-xxiv
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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. Chapman, James H. Charlesworth, Kenneth R. Chase, Chen Zemin, Luciano Chianeque, Philip Chia Phin Yin, Francisca H. Chimhanda, Daniel Chiquete, John T. Chirban, Soobin Choi, Robert Choquette, Mita Choudhury, Gerald Christianson, John Chryssavgis, Sejong Chun, Esther Chung-Kim, Charles M. A. Clark, Elizabeth A. Clark, Sathianathan Clarke, Fred Cloud, John B. Cobb, W. Owen Cole, John A Coleman, John J. Collins, Sylvia Collins-Mayo, Paul K. Conkin, Beth A. Conklin, Sean Connolly, Demetrios J. Constantelos, Michael A. Conway, Paula M. Cooey, Austin Cooper, Michael L. Cooper-White, Pamela Cooper-White, L. William Countryman, Sérgio Coutinho, Pamela Couture, Shannon Craigo-Snell, James L. Crenshaw, David Crowner, Humberto Horacio Cucchetti, Lawrence S. Cunningham, Elizabeth Mason Currier, Emmanuel Cutrone, Mary L. Daniel, David D. Daniels, Robert Darden, Rolf Darge, Isaiah Dau, Jeffry C. Davis, Jane Dawson, Valentin Dedji, John W. de Gruchy, Paul DeHart, Wendy J. Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. Flowers, Carole Fontaine, David Ford, Mary Ford, Stephanie A. Ford, Jim Forest, William Franke, Robert M. Franklin, Ruth Franzén, Edward H. Friedman, Samuel Frouisou, Lorelei F. Fuchs, Jojo M. Fung, Inger Furseth, Richard R. Gaillardetz, Brandon Gallaher, China Galland, Mark Galli, Ismael García, Tharscisse Gatwa, Jean-Marie Gaudeul, Luis María Gavilanes del Castillo, Pavel L. Gavrilyuk, Volney P. Gay, Metropolitan Athanasios Geevargis, Kondothra M. George, Mary Gerhart, Simon Gikandi, Maurice Gilbert, Michael J. Gillgannon, Verónica Giménez Beliveau, Terryl Givens, Beth Glazier-McDonald, Philip Gleason, Menghun Goh, Brian Golding, Bishop Hilario M. Gomez, Michelle A. Gonzalez, Donald K. Gorrell, Roy Gottfried, Tamara Grdzelidze, Joel B. Green, Niels Henrik Gregersen, Cristina Grenholm, Herbert Griffiths, Eric W. Gritsch, Erich S. Gruen, Christoffer H. Grundmann, Paul H. Gundani, Jon P. Gunnemann, Petre Guran, Vidar L. Haanes, Jeremiah M. Hackett, Getatchew Haile, Douglas John Hall, Nicholas Hammond, Daphne Hampson, Jehu J. Hanciles, Barry Hankins, Jennifer Haraguchi, Stanley S. Harakas, Anthony John Harding, Conrad L. Harkins, J. William Harmless, Marjory Harper, Amir Harrak, Joel F. Harrington, Mark W. Harris, Susan Ashbrook Harvey, Van A. Harvey, R. Chris Hassel, Jione Havea, Daniel Hawk, Diana L. Hayes, Leslie Hayes, Priscilla Hayner, S. Mark Heim, Simo Heininen, Richard P. Heitzenrater, Eila Helander, David Hempton, Scott H. Hendrix, Jan-Olav Henriksen, Gina Hens-Piazza, Carter Heyward, Nicholas J. Higham, David Hilliard, Norman A. Hjelm, Peter C. Hodgson, Arthur Holder, M. Jan Holton, Dwight N. Hopkins, Ronnie Po-chia Hsia, Po-Ho Huang, James Hudnut-Beumler, Jennifer S. Hughes, Leonard M. Hummel, Mary E. Hunt, Laennec Hurbon, Mark Hutchinson, Susan E. Hylen, Mary Beth Ingham, H. Larry Ingle, Dale T. Irvin, Jon Isaak, Paul John Isaak, Ada María Isasi-Díaz, Hans Raun Iversen, Margaret C. Jacob, Arthur James, Maria Jansdotter-Samuelsson, David Jasper, Werner G. Jeanrond, Renée Jeffery, David Lyle Jeffrey, Theodore W. Jennings, David H. Jensen, Robin Margaret Jensen, David Jobling, Dale A. Johnson, Elizabeth A. Johnson, Maxwell E. Johnson, Sarah Johnson, Mark D. Johnston, F. Stanley Jones, James William Jones, John R. Jones, Alissa Jones Nelson, Inge Jonsson, Jan Joosten, Elizabeth Judd, Mulambya Peggy Kabonde, Robert Kaggwa, Sylvester Kahakwa, Isaac Kalimi, Ogbu U. Kalu, Eunice Kamaara, Wayne C. Kannaday, Musimbi Kanyoro, Veli-Matti Kärkkäinen, Frank Kaufmann, Léon Nguapitshi Kayongo, Richard Kearney, Alice A. Keefe, Ralph Keen, Catherine Keller, Anthony J. Kelly, Karen Kennelly, Kathi Lynn Kern, Fergus Kerr, Edward Kessler, George Kilcourse, Heup Young Kim, Kim Sung-Hae, Kim Yong-Bock, Kim Yung Suk, Richard King, Thomas M. King, Robert M. Kingdon, Ross Kinsler, Hans G. Kippenberg, Cheryl A. Kirk-Duggan, Clifton Kirkpatrick, Leonid Kishkovsky, Nadieszda Kizenko, Jeffrey Klaiber, Hans-Josef Klauck, Sidney Knight, Samuel Kobia, Robert Kolb, Karla Ann Koll, Heikki Kotila, Donald Kraybill, Philip D. W. Krey, Yves Krumenacker, Jeffrey Kah-Jin Kuan, Simanga R. Kumalo, Peter Kuzmic, Simon Shui-Man Kwan, Kwok Pui-lan, André LaCocque, Stephen E. Lahey, John Tsz Pang Lai, Emiel Lamberts, Armando Lampe, Craig Lampe, Beverly J. Lanzetta, Eve LaPlante, Lizette Larson-Miller, Ariel Bybee Laughton, Leonard Lawlor, Bentley Layton, Robin A. Leaver, Karen Lebacqz, Archie Chi Chung Lee, Marilyn J. Legge, Hervé LeGrand, D. L. LeMahieu, Raymond Lemieux, Bill J. Leonard, Ellen M. Leonard, Outi Leppä, Jean Lesaulnier, Nantawan Boonprasat Lewis, Henrietta Leyser, Alexei Lidov, Bernard Lightman, Paul Chang-Ha Lim, Carter Lindberg, Mark R. Lindsay, James R. Linville, James C. Livingston, Ann Loades, David Loades, Jean-Claude Loba-Mkole, Lo Lung Kwong, Wati Longchar, Eleazar López, David W. Lotz, Andrew Louth, Robin W. Lovin, William Luis, Frank D. Macchia, Diarmaid N. J. MacCulloch, Kirk R. MacGregor, Marjory A. MacLean, Donald MacLeod, Tomas S. Maddela, Inge Mager, Laurenti Magesa, David G. Maillu, Fortunato Mallimaci, Philip Mamalakis, Kä Mana, Ukachukwu Chris Manus, Herbert Robinson Marbury, Reuel Norman Marigza, Jacqueline Mariña, Antti Marjanen, Luiz C. L. Marques, Madipoane Masenya (ngwan'a Mphahlele), Caleb J. D. Maskell, Steve Mason, Thomas Massaro, Fernando Matamoros Ponce, András Máté-Tóth, Odair Pedroso Mateus, Dinis Matsolo, Fumitaka Matsuoka, John D'Arcy May, Yelena Mazour-Matusevich, Theodore Mbazumutima, John S. McClure, Christian McConnell, Lee Martin McDonald, Gary B. McGee, Thomas McGowan, Alister E. McGrath, Richard J. McGregor, John A. McGuckin, Maud Burnett McInerney, Elsie Anne McKee, Mary B. McKinley, James F. McMillan, Ernan McMullin, Kathleen E. McVey, M. 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Phan, Isabel Apawo Phiri, William S. F. Pickering, Derrick G. Pitard, William Elvis Plata, Zlatko Plese, John Plummer, James Newton Poling, Ronald Popivchak, Andrew Porter, Ute Possekel, James M. Powell, Enos Das Pradhan, Devadasan Premnath, Jaime Adrían Prieto Valladares, Anne Primavesi, Randall Prior, María Alicia Puente Lutteroth, Eduardo Guzmão Quadros, Albert Rabil, Laurent William Ramambason, Apolonio M. Ranche, Vololona Randriamanantena Andriamitandrina, Lawrence R. Rast, Paul L. Redditt, Adele Reinhartz, Rolf Rendtorff, Pål Repstad, James N. Rhodes, John K. Riches, Joerg Rieger, Sharon H. Ringe, Sandra Rios, Tyler Roberts, David M. Robinson, James M. Robinson, Joanne Maguire Robinson, Richard A. H. Robinson, Roy R. Robson, Jack B. Rogers, Maria Roginska, Sidney Rooy, Rev. Garnett Roper, Maria José Fontelas Rosado-Nunes, Andrew C. Ross, Stefan Rossbach, François Rossier, John D. Roth, John K. Roth, Phillip Rothwell, Richard E. 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Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
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- The Cambridge Dictionary of Christianity
- Published online:
- 05 August 2012
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- 20 September 2010, pp xi-xliv
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Multiplicity-Free Schubert Calculus
- Hugh Thomas, Alexander Yong
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- Journal:
- Canadian Mathematical Bulletin / Volume 53 / Issue 1 / 01 March 2010
- Published online by Cambridge University Press:
- 20 November 2018, pp. 171-186
- Print publication:
- 01 March 2010
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Multiplicity-free algebraic geometry is the study of subvarieties $Y\,\subseteq \,X$ with the “smallest invariants” as witnessed by a multiplicity-free Chow ring decomposition of $\left[ Y \right]\,\in \,{{A}^{*}}\left( X \right)$ into a predetermined linear basis.
This paper concerns the case of Richardson subvarieties of the Grassmannian in terms of the Schubert basis. We give a nonrecursive combinatorial classification of multiplicity-free Richardson varieties, i.e., we classify multiplicity-free products of Schubert classes. This answers a question of W. Fulton.
Processing and On-Wafer Measurements of Ferroelectric Interdigitated Tunable Microwave Capacitors
- Alexander M. Grishin, Jang-Yong Kim, Sergey I. Khartsev
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- Journal:
- MRS Online Proceedings Library Archive / Volume 811 / 2004
- Published online by Cambridge University Press:
- 28 July 2011, D10.1
- Print publication:
- 2004
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Na0.5K0.5NbO3 (NKN) and Pb(Zr0.53Ti0.47)O3 (PZT) films have been grown by rf-magnetron sputtering and pulsed laser deposition techniques, correspondingly, on sapphire (Al2O3-0112, rcut), YAlO3 + 1% Nd (Nd:YAlO3-001), and quartz (Y+36°-cut) single crystal substrates. Interdigital capacitor (IDC) of coplanar waveguide (CPW) structures were defined by a standard lift off technique in a Au(0.5μm)/Cr(10nm) electrode electron beam evaporated on ferroelectric film surface. IDCs consisted of five pairs of fingers separated by 2 and 4 μm gap. On-wafer microwave characterization was performed using a workbench equipped with a coplanar probe station (Cascade Microtech) with G-S-G (Ground-Signal-Ground) Picoprobe, a network analyzer (Agilent Technologies E8364A) operating in 45 MHz to 40 GHz range and programmable power supply for dc DUT (Device Under Test) biasing. Assumed equivalent circuit for the IDC/CPW structure contains planar capacitor under test C, the coplanar line with a complex impedance ς and a parasitic capacitance Cp between the signal and ground lines. The de-embedding technique has been employed to determine all six complex parameters C, ς and Cp from S-parameter measurements performed for three different device structures: device, open and thru. NKN film interdigital capacitors on sapphire show superior performance in this microwave range: the frequency dispersion was as low as 18%, voltage tunability = 1 – C(40V)/C(0) (40 V, 200 kV/cm) about 14%, loss tangent ∼ 0.11, K-factor = tunability/tanδ from 131% @ 10 GHz to 56% @ 40 GHz. The reliability of the de-embedding procedure is clearly proved by analysis of the frequency dependences of the parasitic capacitance and loss tangent as well as impedance of the coplanar line. Within the accuracy of experimental data and de-embedding calculations these values appear to be voltage independent: Cp ∼ 70 fF, tan δp changes from 0.07@10 GHz to 0.15@40 GHz; real and imaginary part of interconnect impedance increases with frequency from 0.16Ω@10 GHz to 0.36Ω@40 GHz and from 1.6Ω@10 GHz to 5.84Ω@40 GHz respectively.